From 598ac18969ed9b131060327d2213d4fc96b0f6ef Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Fri, 17 Sep 2021 20:59:35 +0200
Subject: [PATCH 01/17] initial commit

---
 src/analysis/convex/combination.lean | 72 ++++++++++++++--------------
 1 file changed, 36 insertions(+), 36 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 7ab43de5178c7..8eca503e06552 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -25,44 +25,44 @@ open set
 open_locale big_operators classical
 
 universes u u'
-variables {E F ι ι' : Type*} [add_comm_group E] [module ℝ E] [add_comm_group F] [module ℝ F]
-  {s : set E}
+variables {𝕜 E ι ι' : Type*} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
 
 /-- Center of mass of a finite collection of points with prescribed weights.
 Note that we require neither `0 ≤ w i` nor `∑ w = 1`. -/
-noncomputable def finset.center_mass (t : finset ι) (w : ι → ℝ) (z : ι → E) : E :=
-(∑ i in t, w i)⁻¹ • (∑ i in t, w i • z i)
+def linear_combination (p : ι → E) :
+  (ι →₀ 𝕜) →ₗ[𝕜] E :=
+⟨λ w : ι →₀ 𝕜, w.sum (λ i _, w i • p i), sorry⟩
 
-variables (i j : ι) (c : ℝ) (t : finset ι) (w : ι → ℝ) (z : ι → E)
+variables (i j : ι) (c : ℝ) (t : finset ι) (w : ι →₀ 𝕜) (p : ι → E)
 
 open finset
 
-lemma finset.center_mass_empty : (∅ : finset ι).center_mass w z = 0 :=
-by simp only [center_mass, sum_empty, smul_zero]
+lemma linear_combination_zero : linear_combination p (0 : ι →₀ 𝕜) = 0 :=
+by rw [linear_combination, linear_map.map_zero]
 
 lemma finset.center_mass_pair (hne : i ≠ j) :
-  ({i, j} : finset ι).center_mass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j :=
+  ({i, j} : finset ι).center_mass w p = (w i / (w i + w j)) • p i + (w j / (w i + w j)) • p j :=
 by simp only [center_mass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul]
 
 variable {w}
 
 lemma finset.center_mass_insert (ha : i ∉ t) (hw : ∑ j in t, w j ≠ 0) :
-  (insert i t).center_mass w z = (w i / (w i + ∑ j in t, w j)) • z i +
-    ((∑ j in t, w j) / (w i + ∑ j in t, w j)) • t.center_mass w z :=
+  (insert i t).center_mass w p = (w i / (w i + ∑ j in t, w j)) • p i +
+    ((∑ j in t, w j) / (w i + ∑ j in t, w j)) • t.center_mass w p :=
 begin
   simp only [center_mass, sum_insert ha, smul_add, (mul_smul _ _ _).symm, ← div_eq_inv_mul],
   congr' 2,
   rw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div]
 end
 
-lemma finset.center_mass_singleton (hw : w i ≠ 0) : ({i} : finset ι).center_mass w z = z i :=
+lemma finset.center_mass_singleton (hw : w i ≠ 0) : ({i} : finset ι).center_mass w p = p i :=
 by rw [center_mass, sum_singleton, sum_singleton, ← mul_smul, inv_mul_cancel hw, one_smul]
 
 lemma finset.center_mass_eq_of_sum_1 (hw : ∑ i in t, w i = 1) :
-  t.center_mass w z = ∑ i in t, w i • z i :=
+  t.center_mass w p = ∑ i in t, w i • p i :=
 by simp only [finset.center_mass, hw, inv_one, one_smul]
 
-lemma finset.center_mass_smul : t.center_mass w (λ i, c • z i) = c • t.center_mass w z :=
+lemma finset.center_mass_smul : t.center_mass w (λ i, c • p i) = c • t.center_mass w p :=
 by simp only [finset.center_mass, finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc]
 
 /-- A convex combination of two centers of mass is a center of mass as well. This version
@@ -84,19 +84,19 @@ end
 /-- A convex combination of two centers of mass is a center of mass as well. This version
 works if two centers of mass share the set of original points. -/
 lemma finset.center_mass_segment
-  (s : finset ι) (w₁ w₂ : ι → ℝ) (z : ι → E)
+  (s : finset ι) (w₁ w₂ : ι → ℝ) (p : ι → E)
   (hw₁ : ∑ i in s, w₁ i = 1) (hw₂ : ∑ i in s, w₂ i = 1) (a b : ℝ) (hab : a + b = 1) :
-  a • s.center_mass w₁ z + b • s.center_mass w₂ z =
-    s.center_mass (λ i, a * w₁ i + b * w₂ i) z :=
+  a • s.center_mass w₁ p + b • s.center_mass w₂ p =
+    s.center_mass (λ i, a * w₁ i + b * w₂ i) p :=
 have hw : ∑ i in s, (a * w₁ i + b * w₂ i) = 1,
   by simp only [mul_sum.symm, sum_add_distrib, mul_one, *],
 by simp only [finset.center_mass_eq_of_sum_1, smul_sum, sum_add_distrib, add_smul, mul_smul, *]
 
 lemma finset.center_mass_ite_eq (hi : i ∈ t) :
-  t.center_mass (λ j, if (i = j) then 1 else 0) z = z i :=
+  t.center_mass (λ j, if (i = j) then 1 else 0) p = p i :=
 begin
   rw [finset.center_mass_eq_of_sum_1],
-  transitivity ∑ j in t, if (i = j) then z i else 0,
+  transitivity ∑ j in t, if (i = j) then p i else 0,
   { congr' with i, split_ifs, exacts [h ▸ one_smul _ _, zero_smul _ _] },
   { rw [sum_ite_eq, if_pos hi] },
   { rw [sum_ite_eq, if_pos hi] }
@@ -106,7 +106,7 @@ variables {t w}
 
 lemma finset.center_mass_subset {t' : finset ι} (ht : t ⊆ t')
   (h : ∀ i ∈ t', i ∉ t → w i = 0) :
-  t.center_mass w z = t'.center_mass w z :=
+  t.center_mass w p = t'.center_mass w p :=
 begin
   rw [center_mass, sum_subset ht h, smul_sum, center_mass, smul_sum],
   apply sum_subset ht,
@@ -115,8 +115,8 @@ begin
 end
 
 lemma finset.center_mass_filter_ne_zero :
-  (t.filter (λ i, w i ≠ 0)).center_mass w z = t.center_mass w z :=
-finset.center_mass_subset z (filter_subset _ _) $ λ i hit hit',
+  (t.filter (λ i, w i ≠ 0)).center_mass w p = t.center_mass w p :=
+finset.center_mass_subset p (filter_subset _ _) $ λ i hit hit',
   by simpa only [hit, mem_filter, true_and, ne.def, not_not] using hit'
 
 variable {z}
@@ -124,16 +124,16 @@ variable {z}
 /-- The center of mass of a finite subset of a convex set belongs to the set
 provided that all weights are non-negative, and the total weight is positive. -/
 lemma convex.center_mass_mem (hs : convex ℝ s) :
-  (∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i in t, w i) → (∀ i ∈ t, z i ∈ s) → t.center_mass w z ∈ s :=
+  (∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i in t, w i) → (∀ i ∈ t, p i ∈ s) → t.center_mass w p ∈ s :=
 begin
   induction t using finset.induction with i t hi ht, { simp [lt_irrefl] },
   intros h₀ hpos hmem,
-  have zi : z i ∈ s, from hmem _ (mem_insert_self _ _),
+  have zi : p i ∈ s, from hmem _ (mem_insert_self _ _),
   have hs₀ : ∀ j ∈ t, 0 ≤ w j, from λ j hj, h₀ j $ mem_insert_of_mem hj,
   rw [sum_insert hi] at hpos,
   by_cases hsum_t : ∑ j in t, w j = 0,
   { have ws : ∀ j ∈ t, w j = 0, from (sum_eq_zero_iff_of_nonneg hs₀).1 hsum_t,
-    have wz : ∑ j in t, w j • z j = 0, from sum_eq_zero (λ i hi, by simp [ws i hi]),
+    have wp : ∑ j in t, w j • p j = 0, from sum_eq_zero (λ i hi, by simp [ws i hi]),
     simp only [center_mass, sum_insert hi, wz, hsum_t, add_zero],
     simp only [hsum_t, add_zero] at hpos,
     rw [← mul_smul, inv_mul_cancel (ne_of_gt hpos), one_smul],
@@ -146,8 +146,8 @@ begin
 end
 
 lemma convex.sum_mem (hs : convex ℝ s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
-  (hz : ∀ i ∈ t, z i ∈ s) :
-  ∑ i in t, w i • z i ∈ s :=
+  (hp : ∀ i ∈ t, p i ∈ s) :
+  ∑ i in t, w i • p i ∈ s :=
 by simpa only [h₁, center_mass, inv_one, one_smul] using
   hs.center_mass_mem h₀ (h₁.symm ▸ zero_lt_one) hz
 
@@ -160,7 +160,7 @@ begin
   intros h x y hx hy a b ha hb hab,
   by_cases h_cases: x = y,
   { rw [h_cases, ←add_smul, hab, one_smul], exact hy },
-  { convert h {x, y} (λ z, if z = y then b else a) _ _ _,
+  { convert h {x, y} (λ z, if p = y then b else a) _ _ _,
     { simp only [sum_pair h_cases, if_neg h_cases, if_pos rfl] },
     { simp_intros i hi,
       cases hi; subst i; simp [ha, hb, if_neg h_cases] },
@@ -170,18 +170,18 @@ begin
 end
 
 lemma finset.center_mass_mem_convex_hull (t : finset ι) {w : ι → ℝ} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
-  (hws : 0 < ∑ i in t, w i) {z : ι → E} (hz : ∀ i ∈ t, z i ∈ s) :
-  t.center_mass w z ∈ convex_hull ℝ s :=
-(convex_convex_hull ℝ s).center_mass_mem hw₀ hws (λ i hi, subset_convex_hull ℝ s $ hz i hi)
+  (hws : 0 < ∑ i in t, w i) {p : ι → E} (hp : ∀ i ∈ t, p i ∈ s) :
+  t.center_mass w p ∈ convex_hull ℝ s :=
+(convex_convex_hull ℝ s).center_mass_mem hw₀ hws (λ i hi, subset_convex_hull ℝ s $ hp i hi)
 
 -- TODO : Do we need other versions of the next lemma?
 
-/-- Convex hull of `s` is equal to the set of all centers of masses of `finset`s `t`, `z '' t ⊆ s`.
+/-- Convex hull of `s` is equal to the set of all centers of masses of `finset`s `t`, `p '' t ⊆ s`.
 This version allows finsets in any type in any universe. -/
 lemma convex_hull_eq (s : set E) :
-  convex_hull ℝ s = {x : E | ∃ (ι : Type u') (t : finset ι) (w : ι → ℝ) (z : ι → E)
-    (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hz : ∀ i ∈ t, z i ∈ s),
-    t.center_mass w z = x} :=
+  convex_hull ℝ s = {x : E | ∃ (ι : Type u') (t : finset ι) (w : ι → ℝ) (p : ι → E)
+    (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hp : ∀ i ∈ t, p i ∈ s),
+    t.center_mass w p = x} :=
 begin
   refine subset.antisymm (convex_hull_min _ _) _,
   { intros x hx,
@@ -202,7 +202,7 @@ begin
       rw [finset.mem_union, finset.mem_map, finset.mem_map] at hi,
       rcases hi with ⟨j, hj, rfl⟩|⟨j, hj, rfl⟩; apply_rules [hzx, hzy] } },
   { rintros _ ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩,
-    exact t.center_mass_mem_convex_hull hw₀ (hw₁.symm ▸ zero_lt_one) hz }
+    exact t.center_mass_mem_convex_hull hw₀ (hw₁.symm ▸ zero_lt_one) hp }
 end
 
 lemma finset.convex_hull_eq (s : finset E) :
@@ -243,7 +243,7 @@ begin
     simp only [mem_Union],
     refine ⟨t.image z, _, _⟩,
     { rw [coe_image, set.image_subset_iff],
-      exact hz },
+      exact hp },
     { apply t.center_mass_mem_convex_hull hw₀,
       { simp only [hw₁, zero_lt_one] },
       { exact λ i hi, finset.mem_coe.2 (finset.mem_image_of_mem _ hi) } } },

From 86ee2142dc0209c362f141c9b22f651c9ada68c9 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Fri, 17 Sep 2021 23:17:25 +0200
Subject: [PATCH 02/17] some more

---
 src/analysis/convex/combination.lean | 233 ++++++++++++++-------------
 1 file changed, 119 insertions(+), 114 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 8eca503e06552..3c4fd73181de4 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -12,148 +12,153 @@ This file defines convex combinations of points in a vector space.
 
 ## Main declarations
 
-* `finset.center_mass`: Center of mass of a finite family of points.
+* `finset.linear_combination`: Center of mass of a finite family of points.
 
 ## Implementation notes
 
-We divide by the sum of the weights in the definition of `finset.center_mass` because of the way
+We divide by the sum of the weights in the definition of `finset.linear_combination` because of the way
 mathematical arguments go: one doesn't change weights, but merely adds some. This also makes a few
 lemmas unconditional on the sum of the weights being `1`.
+
+## TODO
+
+Change `finset.linear_combination : finset ι → (ι → E) → (ι → 𝕜) → E` to
+`linear_combination : (ι → E) →ₗ[𝕜] (ι →₀ 𝕜) →ₗ[𝕜] E`. Same goes for `finset.affine_combination`.
 -/
 
 open set
 open_locale big_operators classical
 
-universes u u'
-variables {𝕜 E ι ι' : Type*} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
+namespace finset
 
-/-- Center of mass of a finite collection of points with prescribed weights.
-Note that we require neither `0 ≤ w i` nor `∑ w = 1`. -/
-def linear_combination (p : ι → E) :
-  (ι →₀ 𝕜) →ₗ[𝕜] E :=
-⟨λ w : ι →₀ 𝕜, w.sum (λ i _, w i • p i), sorry⟩
+/-- Linear combination of a finite collection of points with prescribed weights. -/
+def linear_combination {𝕜 E ι : Type*} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
+  (s : finset ι) (p : ι → E) (w : ι → 𝕜) : E :=
+∑ i in s, w i • p i
 
-variables (i j : ι) (c : ℝ) (t : finset ι) (w : ι →₀ 𝕜) (p : ι → E)
+section ordered_semiring
+variables {𝕜 E ι ι' : Type*} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
+  (i j : ι) (c : 𝕜) (s : finset ι) (p : ι → E) (w : ι → 𝕜)
 
-open finset
+lemma linear_combination_def :
+  s.linear_combination p w = ∑ i in s, w i • p i := rfl
 
-lemma linear_combination_zero : linear_combination p (0 : ι →₀ 𝕜) = 0 :=
-by rw [linear_combination, linear_map.map_zero]
+lemma linear_combination_empty : (∅ : finset ι).linear_combination p w = 0 :=
+by simp only [linear_combination, sum_empty, smul_zero]
 
-lemma finset.center_mass_pair (hne : i ≠ j) :
-  ({i, j} : finset ι).center_mass w p = (w i / (w i + w j)) • p i + (w j / (w i + w j)) • p j :=
-by simp only [center_mass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul]
+lemma linear_combination_pair (hne : i ≠ j) :
+  ({i, j} : finset ι).linear_combination p w = w i • p i + w j • p j :=
+by rw [linear_combination, sum_pair hne]
 
 variable {w}
 
-lemma finset.center_mass_insert (ha : i ∉ t) (hw : ∑ j in t, w j ≠ 0) :
-  (insert i t).center_mass w p = (w i / (w i + ∑ j in t, w j)) • p i +
-    ((∑ j in t, w j) / (w i + ∑ j in t, w j)) • t.center_mass w p :=
-begin
-  simp only [center_mass, sum_insert ha, smul_add, (mul_smul _ _ _).symm, ← div_eq_inv_mul],
-  congr' 2,
-  rw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div]
-end
-
-lemma finset.center_mass_singleton (hw : w i ≠ 0) : ({i} : finset ι).center_mass w p = p i :=
-by rw [center_mass, sum_singleton, sum_singleton, ← mul_smul, inv_mul_cancel hw, one_smul]
-
-lemma finset.center_mass_eq_of_sum_1 (hw : ∑ i in t, w i = 1) :
-  t.center_mass w p = ∑ i in t, w i • p i :=
-by simp only [finset.center_mass, hw, inv_one, one_smul]
+lemma linear_combination_singleton :
+  ({i} : finset ι).linear_combination p w = w i • p i :=
+by rw [linear_combination, sum_singleton]
 
-lemma finset.center_mass_smul : t.center_mass w (λ i, c • p i) = c • t.center_mass w p :=
-by simp only [finset.center_mass, finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc]
+lemma linear_combination_insert (ha : i ∉ s) :
+  (insert i s).linear_combination p w = w i • p i + s.linear_combination p w :=
+by rw [linear_combination, linear_combination, sum_insert ha]
 
 /-- A convex combination of two centers of mass is a center of mass as well. This version
 deals with two different index types. -/
-lemma finset.center_mass_segment'
-  (s : finset ι) (t : finset ι') (ws : ι → ℝ) (zs : ι → E) (wt : ι' → ℝ) (zt : ι' → E)
-  (hws : ∑ i in s, ws i = 1) (hwt : ∑ i in t, wt i = 1) (a b : ℝ) (hab : a + b = 1) :
-  a • s.center_mass ws zs + b • t.center_mass wt zt =
-    (s.map function.embedding.inl ∪ t.map function.embedding.inr).center_mass
-      (sum.elim (λ i, a * ws i) (λ j, b * wt j))
-      (sum.elim zs zt) :=
+lemma linear_combination_segment' (s : finset ι) (t : finset ι') (ws : ι → 𝕜) (ps : ι → E)
+  (wt : ι' → 𝕜) (pt : ι' → E) (a b : 𝕜) (hab : a + b = 1) :
+  a • s.linear_combination ps ws + b • t.linear_combination pt wt =
+    (s.map function.embedding.inl ∪ t.map function.embedding.inr).linear_combination
+      (sum.elim ps pt)
+      (sum.elim (λ i, a * ws i) (λ j, b * wt j)) :=
 begin
-  rw [s.center_mass_eq_of_sum_1 _ hws, t.center_mass_eq_of_sum_1 _ hwt,
-    smul_sum, smul_sum, ← finset.sum_sum_elim, finset.center_mass_eq_of_sum_1],
-  { congr' with ⟨⟩; simp only [sum.elim_inl, sum.elim_inr, mul_smul] },
-  { rw [sum_sum_elim, ← mul_sum, ← mul_sum, hws, hwt, mul_one, mul_one, hab] }
+  unfold linear_combination,
+  rw [smul_sum, smul_sum, ← sum_sum_elim],
+  { congr' with ⟨⟩; simp only [sum.elim_inl, sum.elim_inr, mul_smul] }
 end
 
 /-- A convex combination of two centers of mass is a center of mass as well. This version
 works if two centers of mass share the set of original points. -/
-lemma finset.center_mass_segment
-  (s : finset ι) (w₁ w₂ : ι → ℝ) (p : ι → E)
-  (hw₁ : ∑ i in s, w₁ i = 1) (hw₂ : ∑ i in s, w₂ i = 1) (a b : ℝ) (hab : a + b = 1) :
-  a • s.center_mass w₁ p + b • s.center_mass w₂ p =
-    s.center_mass (λ i, a * w₁ i + b * w₂ i) p :=
-have hw : ∑ i in s, (a * w₁ i + b * w₂ i) = 1,
-  by simp only [mul_sum.symm, sum_add_distrib, mul_one, *],
-by simp only [finset.center_mass_eq_of_sum_1, smul_sum, sum_add_distrib, add_smul, mul_smul, *]
-
-lemma finset.center_mass_ite_eq (hi : i ∈ t) :
-  t.center_mass (λ j, if (i = j) then 1 else 0) p = p i :=
+lemma linear_combination_segment (s : finset ι) (w₁ w₂ : ι → 𝕜) (p : ι → E) (a b : 𝕜)
+  (hab : a + b = 1) :
+  a • s.linear_combination p w₁ + b • s.linear_combination p w₂ =
+    s.linear_combination p (λ i, a * w₁ i + b * w₂ i) :=
 begin
-  rw [finset.center_mass_eq_of_sum_1],
-  transitivity ∑ j in t, if (i = j) then p i else 0,
+  unfold linear_combination,
+  simp only [linear_combination_def, smul_sum, sum_add_distrib, add_smul, mul_smul, *],
+end
+
+lemma linear_combination_ite_eq (hi : i ∈ s) :
+  s.linear_combination p (λ j, if (i = j) then 1 else 0) = p i :=
+begin
+  rw linear_combination,
+  transitivity ∑ j in s, if (i = j) then p i else 0,
   { congr' with i, split_ifs, exacts [h ▸ one_smul _ _, zero_smul _ _] },
-  { rw [sum_ite_eq, if_pos hi] },
   { rw [sum_ite_eq, if_pos hi] }
 end
 
-variables {t w}
+variables {s w}
 
-lemma finset.center_mass_subset {t' : finset ι} (ht : t ⊆ t')
-  (h : ∀ i ∈ t', i ∉ t → w i = 0) :
-  t.center_mass w p = t'.center_mass w p :=
+lemma linear_combination_subset {t : finset ι} (ht : s ⊆ t)
+  (h : ∀ i ∈ t, i ∉ s → w i = 0) :
+  s.linear_combination p w = t.linear_combination p w :=
 begin
-  rw [center_mass, sum_subset ht h, smul_sum, center_mass, smul_sum],
-  apply sum_subset ht,
-  assume i hit' hit,
-  rw [h i hit' hit, zero_smul, smul_zero]
+  rw [linear_combination, linear_combination],
+  exact sum_subset ht (λ i hit his, by rw [h i hit his, zero_smul]),
 end
 
-lemma finset.center_mass_filter_ne_zero :
-  (t.filter (λ i, w i ≠ 0)).center_mass w p = t.center_mass w p :=
-finset.center_mass_subset p (filter_subset _ _) $ λ i hit hit',
+lemma linear_combination_filter_ne_zero :
+  (s.filter (λ i, w i ≠ 0)).linear_combination p w = s.linear_combination p w :=
+linear_combination_subset p (filter_subset _ _) $ λ i hit hit',
   by simpa only [hit, mem_filter, true_and, ne.def, not_not] using hit'
 
-variable {z}
+variables {p} {t : set E}
+
+end ordered_semiring
+
+section ordered_comm_semiring
+variables {𝕜 E ι ι' : Type*} [ordered_comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
+  (c : 𝕜) (s : finset ι) (p : ι → E) {w : ι → 𝕜}
+
+lemma linear_combination_smul :
+  s.linear_combination (λ i, c • p i) w = c • s.linear_combination p w :=
+by simp_rw [linear_combination, smul_sum, smul_comm c]
+
+end ordered_comm_semiring
+
+section linear_ordered_field
+variables {𝕜 E ι ι' : Type*} [linear_ordered_field 𝕜] [add_comm_monoid E] [module 𝕜 E]
+  {s : set E} {t : finset ι} {p : ι → E} {w : ι → 𝕜}
 
 /-- The center of mass of a finite subset of a convex set belongs to the set
 provided that all weights are non-negative, and the total weight is positive. -/
-lemma convex.center_mass_mem (hs : convex ℝ s) :
-  (∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i in t, w i) → (∀ i ∈ t, p i ∈ s) → t.center_mass w p ∈ s :=
+lemma convex.linear_combination_mem (ht : convex 𝕜 s) :
+  (∀ i ∈ t, 0 ≤ w i) → (∑ i in t, w i = 1) → (∀ i ∈ t, p i ∈ s) → t.linear_combination p w ∈ s :=
 begin
   induction t using finset.induction with i t hi ht, { simp [lt_irrefl] },
-  intros h₀ hpos hmem,
+  intros h₀ h₁ hmem,
   have zi : p i ∈ s, from hmem _ (mem_insert_self _ _),
   have hs₀ : ∀ j ∈ t, 0 ≤ w j, from λ j hj, h₀ j $ mem_insert_of_mem hj,
-  rw [sum_insert hi] at hpos,
+  rw [sum_insert hi] at h₁,
+  rw linear_combination_insert _ _ _ hi,
   by_cases hsum_t : ∑ j in t, w j = 0,
-  { have ws : ∀ j ∈ t, w j = 0, from (sum_eq_zero_iff_of_nonneg hs₀).1 hsum_t,
-    have wp : ∑ j in t, w j • p j = 0, from sum_eq_zero (λ i hi, by simp [ws i hi]),
-    simp only [center_mass, sum_insert hi, wz, hsum_t, add_zero],
-    simp only [hsum_t, add_zero] at hpos,
-    rw [← mul_smul, inv_mul_cancel (ne_of_gt hpos), one_smul],
+  { have wt : ∀ j ∈ t, w j = 0, from (sum_eq_zero_iff_of_nonneg hs₀).1 hsum_t,
+    have wp : t.linear_combination p w = 0, from sum_eq_zero (λ i hi, by simp [wt i hi]),
+    rw [hsum_t, add_zero] at h₁,
+    rw [wp, add_zero, h₁, one_smul],
     exact zi },
-  { rw [finset.center_mass_insert _ _ _ hi hsum_t],
-    refine convex_iff_div.1 hs zi (ht hs₀ _ _) _ (sum_nonneg hs₀) hpos,
+  { refine convex_iff_div.1 hs zi (ht hs₀ _ _) _ (sum_nonneg hs₀) h₁,
     { exact lt_of_le_of_ne (sum_nonneg hs₀) (ne.symm hsum_t) },
     { intros j hj, exact hmem j (mem_insert_of_mem hj) },
     { exact h₀ _ (mem_insert_self _ _) } }
 end
 
-lemma convex.sum_mem (hs : convex ℝ s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
+lemma convex.sum_mem (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
   (hp : ∀ i ∈ t, p i ∈ s) :
   ∑ i in t, w i • p i ∈ s :=
-by simpa only [h₁, center_mass, inv_one, one_smul] using
-  hs.center_mass_mem h₀ (h₁.symm ▸ zero_lt_one) hz
+by simpa only [h₁, linear_combination, inv_one, one_smul] using
+  hs.linear_combination_mem h₀ (h₁.symm ▸ zero_lt_one) hz
 
 lemma convex_iff_sum_mem :
-  convex ℝ s ↔
-    (∀ (t : finset E) (w : E → ℝ),
+  convex 𝕜 s ↔
+    (∀ (t : finset E) (w : E → 𝕜),
       (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s ) :=
 begin
   refine ⟨λ hs t w hw₀ hw₁ hts, hs.sum_mem hw₀ hw₁ hts, _⟩,
@@ -169,28 +174,28 @@ begin
       cases hi; subst i; simp [hx, hy, if_neg h_cases] } }
 end
 
-lemma finset.center_mass_mem_convex_hull (t : finset ι) {w : ι → ℝ} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
-  (hws : 0 < ∑ i in t, w i) {p : ι → E} (hp : ∀ i ∈ t, p i ∈ s) :
-  t.center_mass w p ∈ convex_hull ℝ s :=
-(convex_convex_hull ℝ s).center_mass_mem hw₀ hws (λ i hi, subset_convex_hull ℝ s $ hp i hi)
+lemma finset.linear_combination_mem_convex_hull (t : finset ι) {w : ι → 𝕜} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
+  (hws : ∑ i in t, w i = 1) {p : ι → E} (hp : ∀ i ∈ t, p i ∈ s) :
+  s.linear_combination p w ∈ convex_hull 𝕜 s :=
+(convex_convex_hull 𝕜 s).linear_combination_mem hw₀ hws (λ i hi, subset_convex_hull 𝕜 s $ hp i hi)
 
 -- TODO : Do we need other versions of the next lemma?
 
 /-- Convex hull of `s` is equal to the set of all centers of masses of `finset`s `t`, `p '' t ⊆ s`.
 This version allows finsets in any type in any universe. -/
 lemma convex_hull_eq (s : set E) :
-  convex_hull ℝ s = {x : E | ∃ (ι : Type u') (t : finset ι) (w : ι → ℝ) (p : ι → E)
+  convex_hull 𝕜 s = {x : E | ∃ (ι : Type u') (t : finset ι) (w : ι → 𝕜) (p : ι → E)
     (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hp : ∀ i ∈ t, p i ∈ s),
-    t.center_mass w p = x} :=
+    s.linear_combination p w = x} :=
 begin
   refine subset.antisymm (convex_hull_min _ _) _,
   { intros x hx,
     use [punit, {punit.star}, λ _, 1, λ _, x, λ _ _, zero_le_one,
       finset.sum_singleton, λ _ _, hx],
-    simp only [finset.center_mass, finset.sum_singleton, inv_one, one_smul] },
+    simp only [finset.linear_combination, finset.sum_singleton, inv_one, one_smul] },
   { rintros x y ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ ⟨ι', sy, wy, zy, hwy₀, hwy₁, hzy, rfl⟩
       a b ha hb hab,
-    rw [finset.center_mass_segment' _ _ _ _ _ _ hwx₁ hwy₁ _ _ hab],
+    rw [finset.linear_combination_segment' _ _ _ _ _ _ hwx₁ hwy₁ _ _ hab],
     refine ⟨_, _, _, _, _, _, _, rfl⟩,
     { rintros i hi,
       rw [finset.mem_union, finset.mem_map, finset.mem_map] at hi,
@@ -202,40 +207,40 @@ begin
       rw [finset.mem_union, finset.mem_map, finset.mem_map] at hi,
       rcases hi with ⟨j, hj, rfl⟩|⟨j, hj, rfl⟩; apply_rules [hzx, hzy] } },
   { rintros _ ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩,
-    exact t.center_mass_mem_convex_hull hw₀ (hw₁.symm ▸ zero_lt_one) hp }
+    exact s.linear_combination_mem_convex_hull hw₀ (hw₁.symm ▸ zero_lt_one) hp }
 end
 
 lemma finset.convex_hull_eq (s : finset E) :
-  convex_hull ℝ ↑s = {x : E | ∃ (w : E → ℝ) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : ∑ y in s, w y = 1),
-    s.center_mass w id = x} :=
+  convex_hull 𝕜 ↑s = {x : E | ∃ (w : E → 𝕜) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : ∑ y in s, w y = 1),
+    s.linear_combination w id = x} :=
 begin
   refine subset.antisymm (convex_hull_min _ _) _,
   { intros x hx,
     rw [finset.mem_coe] at hx,
-    refine ⟨_, _, _, finset.center_mass_ite_eq _ _ _ hx⟩,
+    refine ⟨_, _, _, finset.linear_combination_ite_eq _ _ _ hx⟩,
     { intros, split_ifs, exacts [zero_le_one, le_refl 0] },
     { rw [finset.sum_ite_eq, if_pos hx] } },
   { rintros x y ⟨wx, hwx₀, hwx₁, rfl⟩ ⟨wy, hwy₀, hwy₁, rfl⟩
       a b ha hb hab,
-    rw [finset.center_mass_segment _ _ _ _ hwx₁ hwy₁ _ _ hab],
+    rw [finset.linear_combination_segment _ _ _ _ hwx₁ hwy₁ _ _ hab],
     refine ⟨_, _, _, rfl⟩,
     { rintros i hi,
       apply_rules [add_nonneg, mul_nonneg, hwx₀, hwy₀], },
     { simp only [finset.sum_add_distrib, finset.mul_sum.symm, mul_one, *] } },
   { rintros _ ⟨w, hw₀, hw₁, rfl⟩,
-    exact s.center_mass_mem_convex_hull (λ x hx, hw₀ _ hx)
+    exact s.linear_combination_mem_convex_hull (λ x hx, hw₀ _ hx)
       (hw₁.symm ▸ zero_lt_one) (λ x hx, hx) }
 end
 
 lemma set.finite.convex_hull_eq {s : set E} (hs : finite s) :
-  convex_hull ℝ s = {x : E | ∃ (w : E → ℝ) (hw₀ : ∀ y ∈ s, 0 ≤ w y)
-    (hw₁ : ∑ y in hs.to_finset, w y = 1), hs.to_finset.center_mass w id = x} :=
+  convex_hull 𝕜 s = {x : E | ∃ (w : E → 𝕜) (hw₀ : ∀ y ∈ s, 0 ≤ w y)
+    (hw₁ : ∑ y in hs.to_finset, w y = 1), hs.to_finset.linear_combination w id = x} :=
 by simpa only [set.finite.coe_to_finset, set.finite.mem_to_finset, exists_prop]
   using hs.to_finset.convex_hull_eq
 
 /-- A weak version of Carathéodory's theorem. -/
 lemma convex_hull_eq_union_convex_hull_finite_subsets (s : set E) :
-  convex_hull ℝ s = ⋃ (t : finset E) (w : ↑t ⊆ s), convex_hull ℝ ↑t :=
+  convex_hull 𝕜 s = ⋃ (t : finset E) (w : ↑t ⊆ s), convex_hull 𝕜 ↑t :=
 begin
   refine subset.antisymm _ _,
   { rw convex_hull_eq,
@@ -244,7 +249,7 @@ begin
     refine ⟨t.image z, _, _⟩,
     { rw [coe_image, set.image_subset_iff],
       exact hp },
-    { apply t.center_mass_mem_convex_hull hw₀,
+    { apply s.linear_combination_mem_convex_hull hw₀,
       { simp only [hw₁, zero_lt_one] },
       { exact λ i hi, finset.mem_coe.2 (finset.mem_image_of_mem _ hi) } } },
    { exact Union_subset (λ i, Union_subset convex_hull_mono), },
@@ -252,32 +257,32 @@ end
 
 /-! ### `std_simplex` -/
 
-variables (ι) [fintype ι] {f : ι → ℝ}
+variables (ι) [fintype ι] {f : ι → 𝕜}
 
-/-- `std_simplex ι` is the convex hull of the canonical basis in `ι → ℝ`. -/
+/-- `std_simplex ι` is the convex hull of the canonical basis in `ι → 𝕜`. -/
 lemma convex_hull_basis_eq_std_simplex :
-  convex_hull ℝ (range $ λ(i j:ι), if i = j then (1:ℝ) else 0) = std_simplex ι :=
+  convex_hull 𝕜 (range $ λ(i j:ι), if i = j then (1:𝕜) else 0) = std_simplex ι :=
 begin
   refine subset.antisymm (convex_hull_min _ (convex_std_simplex ι)) _,
   { rintros _ ⟨i, rfl⟩,
     exact ite_eq_mem_std_simplex i },
   { rintros w ⟨hw₀, hw₁⟩,
-    rw [pi_eq_sum_univ w, ← finset.univ.center_mass_eq_of_sum_1 _ hw₁],
-    exact finset.univ.center_mass_mem_convex_hull (λ i hi, hw₀ i)
+    rw [pi_eq_sum_univ w, ← finset.univ.linear_combination_eq_of_sum_1 _ hw₁],
+    exact finset.univ.linear_combination_mem_convex_hull (λ i hi, hw₀ i)
       (hw₁.symm ▸ zero_lt_one) (λ i hi, mem_range_self i) }
 end
 
 variable {ι}
 
-/-- The convex hull of a finite set is the image of the standard simplex in `s → ℝ`
+/-- The convex hull of a finite set is the image of the standard simplex in `s → 𝕜`
 under the linear map sending each function `w` to `∑ x in s, w x • x`.
 
 Since we have no sums over finite sets, we use sum over `@finset.univ _ hs.fintype`.
-The map is defined in terms of operations on `(s → ℝ) →ₗ[ℝ] ℝ` so that later we will not need
+The map is defined in terms of operations on `(s → 𝕜) →ₗ[𝕜] 𝕜` so that later we will not need
 to prove that this map is linear. -/
 lemma set.finite.convex_hull_eq_image {s : set E} (hs : finite s) :
-  convex_hull ℝ s = by haveI := hs.fintype; exact
-    (⇑(∑ x : s, (@linear_map.proj ℝ s _ (λ i, ℝ) _ _ x).smul_right x.1)) '' (std_simplex s) :=
+  convex_hull 𝕜 s = by haveI := hs.fintype; exact
+    (⇑(∑ x : s, (@linear_map.proj 𝕜 s _ (λ i, 𝕜) _ _ x).smul_right x.1)) '' (std_simplex s) :=
 begin
   rw [← convex_hull_basis_eq_std_simplex, ← linear_map.convex_hull_image, ← set.range_comp, (∘)],
   apply congr_arg,
@@ -288,5 +293,5 @@ end
 
 /-- All values of a function `f ∈ std_simplex ι` belong to `[0, 1]`. -/
 lemma mem_Icc_of_mem_std_simplex (hf : f ∈ std_simplex ι) (x) :
-  f x ∈ Icc (0 : ℝ) 1 :=
+  f x ∈ Icc (0 : 𝕜) 1 :=
 ⟨hf.1 x, hf.2 ▸ finset.single_le_sum (λ y hy, hf.1 y) (finset.mem_univ x)⟩

From 67ba2fa8e305b5e85501981302d461ab4760032f Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Sat, 18 Sep 2021 11:03:52 +0200
Subject: [PATCH 03/17] generalize std_simplex

---
 src/analysis/convex/basic.lean        |  19 +--
 src/analysis/convex/caratheodory.lean |   8 +-
 src/analysis/convex/combination.lean  | 166 +++++++++++++++-----------
 src/analysis/convex/function.lean     |  76 ++++++------
 src/analysis/convex/independent.lean  |   2 +-
 src/analysis/convex/integral.lean     |  16 +--
 src/analysis/convex/topology.lean     |  18 +--
 7 files changed, 164 insertions(+), 141 deletions(-)

diff --git a/src/analysis/convex/basic.lean b/src/analysis/convex/basic.lean
index 262e75d603907..cc0923ae048f4 100644
--- a/src/analysis/convex/basic.lean
+++ b/src/analysis/convex/basic.lean
@@ -18,7 +18,7 @@ In a 𝕜-vector space, we define the following objects and properties.
 * `convex_hull 𝕜 s`: The minimal convex set that includes `s`. In order theory speak, this is a
   closure operator.
 * Standard simplex `std_simplex ι [fintype ι]` is the intersection of the positive quadrant with
-  the hyperplane `s.sum = 1` in the space `ι → ℝ`.
+  the hyperplane `s.sum = 1` in the space `ι → 𝕜`.
 
 We also provide various equivalent versions of the definitions above, prove that some specific sets
 are convex.
@@ -1056,24 +1056,25 @@ end ordered_ring
 end convex_hull
 
 
-variables {ι ι' : Type*} [add_comm_group E] [module ℝ E] [add_comm_group F] [module ℝ F] {s : set E}
+variables {ι ι' : Type*} [ordered_semiring 𝕜] [add_comm_group E] [module 𝕜 E] {s : set E}
 
 /-! ### Simplex -/
 
 section simplex
 
-variables (ι) [fintype ι] {f : ι → ℝ}
+variables (ι) [fintype ι] {f : ι → 𝕜}
 
-/-- The standard simplex in the space of functions `ι → ℝ` is the set
-of vectors with non-negative coordinates with total sum `1`. -/
-def std_simplex (ι : Type*) [fintype ι] : set (ι → ℝ) :=
+/-- The standard simplex in the space of finitely supported functions `ι →₀ 𝕜` is the set
+of vectors with non-negative coordinates with total sum `1`. This is the free object in the category
+of convex spaces.-/
+def std_simplex : set (ι → 𝕜) :=
 {f | (∀ x, 0 ≤ f x) ∧ ∑ x, f x = 1}
 
 lemma std_simplex_eq_inter :
-  std_simplex ι = (⋂ x, {f | 0 ≤ f x}) ∩ {f | ∑ x, f x = 1} :=
+  std_simplex 𝕜 ι = (⋂ x, {f | 0 ≤ f x}) ∩ {f | ∑ x, f x = 1} :=
 by { ext f, simp only [std_simplex, set.mem_inter_eq, set.mem_Inter, set.mem_set_of_eq] }
 
-lemma convex_std_simplex : convex ℝ (std_simplex ι) :=
+lemma convex_std_simplex : convex 𝕜 (std_simplex 𝕜 ι) :=
 begin
   refine λ f g hf hg a b ha hb hab, ⟨λ x, _, _⟩,
   { apply_rules [add_nonneg, mul_nonneg, hf.1, hg.1] },
@@ -1084,7 +1085,7 @@ end
 
 variable {ι}
 
-lemma ite_eq_mem_std_simplex (i : ι) : (λ j, ite (i = j) (1:ℝ) 0) ∈ std_simplex ι :=
+lemma ite_eq_mem_std_simplex (i : ι) : (λ j, ite (i = j) (1:𝕜) 0) ∈ std_simplex 𝕜 ι :=
 ⟨λ j, by simp only; split_ifs; norm_num, by rw [finset.sum_ite_eq, if_pos (finset.mem_univ _)]⟩
 
 end simplex
diff --git a/src/analysis/convex/caratheodory.lean b/src/analysis/convex/caratheodory.lean
index c5159e37c09c3..7f1da0be75e7d 100644
--- a/src/analysis/convex/caratheodory.lean
+++ b/src/analysis/convex/caratheodory.lean
@@ -80,12 +80,12 @@ begin
          ... ≤ f e : fpos e het,
       { apply div_nonneg (fpos i₀ (mem_of_subset (filter_subset _ t) mem)) (le_of_lt hg) },
       { simpa only [mem_filter, het, true_and, not_lt] using hes } } },
-  { simp only [subtype.coe_mk, center_mass_eq_of_sum_1 _ id ksum, id],
+  { simp only [subtype.coe_mk, id],
     calc ∑ e in t.erase i₀, k e • e = ∑ e in t, k e • e : sum_erase _ (by rw [hk, zero_smul])
     ... = ∑ e in t, (f e - f i₀ / g i₀ * g e) • e : rfl
-    ... = t.center_mass f id : _,
+    ... = t.linear_combination id f : _,
     simp only [sub_smul, mul_smul, sum_sub_distrib, ← smul_sum, gcombo, smul_zero,
-      sub_zero, center_mass, fsum, inv_one, one_smul, id.def] }
+      sub_zero, linear_combination, fsum, inv_one, one_smul, id.def] }
 end
 
 variables {s : set E} {x : E} (hx : x ∈ convex_hull ℝ s)
@@ -176,6 +176,6 @@ begin
   { erw [finset.sum_attach, finset.sum_filter_ne_zero, hw₂] },
   { change ∑ (i : t') in t'.attach, (λ e, w e • e) ↑i = x,
     erw [finset.sum_attach, finset.sum_filter_of_ne],
-    { rw t.center_mass_eq_of_sum_1 id hw₂ at hw₃, exact hw₃ },
+    { exact hw₃ },
     { intros e he hwe contra, apply hwe, rw [contra, zero_smul] } }
 end
diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 3c4fd73181de4..b4f9a28adca49 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -6,24 +6,23 @@ Authors: Yury Kudriashov
 import analysis.convex.basic
 
 /-!
-# Convex combinations
+# Linear combinations
 
-This file defines convex combinations of points in a vector space.
+This file defines linear combinations of points in a semimodule and proves results about convex
+combinations. Convex combinations in a semimodule can be modelled as linear combinations with
+nonnegative coefficients summing to `1`. In an affine space, they can be modelled through affine combinations.
 
 ## Main declarations
 
 * `finset.linear_combination`: Center of mass of a finite family of points.
 
-## Implementation notes
-
-We divide by the sum of the weights in the definition of `finset.linear_combination` because of the way
-mathematical arguments go: one doesn't change weights, but merely adds some. This also makes a few
-lemmas unconditional on the sum of the weights being `1`.
-
 ## TODO
 
 Change `finset.linear_combination : finset ι → (ι → E) → (ι → 𝕜) → E` to
 `linear_combination : (ι → E) →ₗ[𝕜] (ι →₀ 𝕜) →ₗ[𝕜] E`. Same goes for `finset.affine_combination`.
+
+Ultimately, this file should be about `convex_combination`, which will generalize both
+`affine_combination` and `linear_combination`. The latter should find a home in `linear_algebra`.
 -/
 
 open set
@@ -60,6 +59,10 @@ lemma linear_combination_insert (ha : i ∉ s) :
   (insert i s).linear_combination p w = w i • p i + s.linear_combination p w :=
 by rw [linear_combination, linear_combination, sum_insert ha]
 
+lemma linear_combination_const_left (x : E) :
+  s.linear_combination (λ _, x) w = (∑ i in s, w i) • x :=
+by rw [linear_combination, finset.sum_smul]
+
 /-- A convex combination of two centers of mass is a center of mass as well. This version
 deals with two different index types. -/
 lemma linear_combination_segment' (s : finset ι) (t : finset ι') (ws : ι → 𝕜) (ps : ι → E)
@@ -86,7 +89,7 @@ begin
 end
 
 lemma linear_combination_ite_eq (hi : i ∈ s) :
-  s.linear_combination p (λ j, if (i = j) then 1 else 0) = p i :=
+  s.linear_combination p (λ j, if (i = j) then (1 : 𝕜) else 0) = p i :=
 begin
   rw linear_combination,
   transitivity ∑ j in s, if (i = j) then p i else 0,
@@ -94,6 +97,10 @@ begin
   { rw [sum_ite_eq, if_pos hi] }
 end
 
+lemma linear_combination_smul_right :
+  s.linear_combination p (c • w) = c • s.linear_combination p w :=
+by simp_rw [linear_combination, smul_sum, pi.smul_apply, smul_assoc]
+
 variables {s w}
 
 lemma linear_combination_subset {t : finset ι} (ht : s ⊆ t)
@@ -117,9 +124,9 @@ section ordered_comm_semiring
 variables {𝕜 E ι ι' : Type*} [ordered_comm_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
   (c : 𝕜) (s : finset ι) (p : ι → E) {w : ι → 𝕜}
 
-lemma linear_combination_smul :
-  s.linear_combination (λ i, c • p i) w = c • s.linear_combination p w :=
-by simp_rw [linear_combination, smul_sum, smul_comm c]
+lemma linear_combination_smul_left :
+  s.linear_combination (c • p) w = c • s.linear_combination p w :=
+by simp_rw [linear_combination, smul_sum, pi.smul_apply, smul_comm c]
 
 end ordered_comm_semiring
 
@@ -127,45 +134,52 @@ section linear_ordered_field
 variables {𝕜 E ι ι' : Type*} [linear_ordered_field 𝕜] [add_comm_monoid E] [module 𝕜 E]
   {s : set E} {t : finset ι} {p : ι → E} {w : ι → 𝕜}
 
-/-- The center of mass of a finite subset of a convex set belongs to the set
-provided that all weights are non-negative, and the total weight is positive. -/
-lemma convex.linear_combination_mem (ht : convex 𝕜 s) :
+lemma linear_combination_normalize  (hw : ∑ i in t, w i ≠ 0) :
+  t.linear_combination p w = (∑ i in t, w i) • t.linear_combination p ((∑ i in t, w i)⁻¹ • w) :=
+by rw [linear_combination_smul_right, smul_inv_smul' hw]
+
+/-- The linear combination of a finite subset of a convex set belongs to the set
+provided that all weights are non-negative, and the total weight is `1`. -/
+lemma _root_.convex.linear_combination_mem (hs : convex 𝕜 s) :
   (∀ i ∈ t, 0 ≤ w i) → (∑ i in t, w i = 1) → (∀ i ∈ t, p i ∈ s) → t.linear_combination p w ∈ s :=
 begin
-  induction t using finset.induction with i t hi ht, { simp [lt_irrefl] },
+  induction t using finset.induction with i t hi ht generalizing w, { simp [lt_irrefl] },
   intros h₀ h₁ hmem,
-  have zi : p i ∈ s, from hmem _ (mem_insert_self _ _),
-  have hs₀ : ∀ j ∈ t, 0 ≤ w j, from λ j hj, h₀ j $ mem_insert_of_mem hj,
+  have hpi : p i ∈ s, from hmem _ (mem_insert_self _ _),
+  have ht₀ : ∀ j ∈ t, 0 ≤ w j, from λ j hj, h₀ j $ mem_insert_of_mem hj,
   rw [sum_insert hi] at h₁,
   rw linear_combination_insert _ _ _ hi,
   by_cases hsum_t : ∑ j in t, w j = 0,
-  { have wt : ∀ j ∈ t, w j = 0, from (sum_eq_zero_iff_of_nonneg hs₀).1 hsum_t,
+  { have wt : ∀ j ∈ t, w j = 0, from (sum_eq_zero_iff_of_nonneg ht₀).1 hsum_t,
     have wp : t.linear_combination p w = 0, from sum_eq_zero (λ i hi, by simp [wt i hi]),
     rw [hsum_t, add_zero] at h₁,
     rw [wp, add_zero, h₁, one_smul],
-    exact zi },
-  { refine convex_iff_div.1 hs zi (ht hs₀ _ _) _ (sum_nonneg hs₀) h₁,
-    { exact lt_of_le_of_ne (sum_nonneg hs₀) (ne.symm hsum_t) },
-    { intros j hj, exact hmem j (mem_insert_of_mem hj) },
-    { exact h₀ _ (mem_insert_self _ _) } }
+    exact hpi },
+  rw linear_combination_normalize hsum_t,
+  refine hs hpi _ (h₀ _ (mem_insert_self _ _)) (sum_nonneg ht₀) h₁,
+  refine ht (λ j hj, _) _ (λ j hj, hmem _ (mem_insert_of_mem hj)),
+  { dsimp,
+    refine mul_nonneg (inv_nonneg.2 (sum_nonneg ht₀)) (ht₀ j hj) },
+  dsimp,
+  simp, --nonterminal simp to fix
+  rw [←mul_sum, inv_eq_one_div, one_div_mul_cancel hsum_t],
 end
 
-lemma convex.sum_mem (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
+lemma _root_.convex.sum_mem (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
   (hp : ∀ i ∈ t, p i ∈ s) :
   ∑ i in t, w i • p i ∈ s :=
-by simpa only [h₁, linear_combination, inv_one, one_smul] using
-  hs.linear_combination_mem h₀ (h₁.symm ▸ zero_lt_one) hz
+hs.linear_combination_mem h₀ h₁ hp
 
-lemma convex_iff_sum_mem :
+lemma _root_.convex_iff_linear_combination_mem :
   convex 𝕜 s ↔
     (∀ (t : finset E) (w : E → 𝕜),
-      (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s ) :=
+      (∀ i ∈ t, 0 ≤ w i) → ∑ i in t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x in t, w x • x ∈ s) :=
 begin
-  refine ⟨λ hs t w hw₀ hw₁ hts, hs.sum_mem hw₀ hw₁ hts, _⟩,
+  refine ⟨λ hs t w hw₀ hw₁ hts, hs.linear_combination_mem hw₀ hw₁ hts, _⟩,
   intros h x y hx hy a b ha hb hab,
   by_cases h_cases: x = y,
   { rw [h_cases, ←add_smul, hab, one_smul], exact hy },
-  { convert h {x, y} (λ z, if p = y then b else a) _ _ _,
+  { convert h {x, y} (λ p, if p = y then b else a) _ _ _,
     { simp only [sum_pair h_cases, if_neg h_cases, if_pos rfl] },
     { simp_intros i hi,
       cases hi; subst i; simp [ha, hb, if_neg h_cases] },
@@ -174,105 +188,111 @@ begin
       cases hi; subst i; simp [hx, hy, if_neg h_cases] } }
 end
 
-lemma finset.linear_combination_mem_convex_hull (t : finset ι) {w : ι → 𝕜} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
-  (hws : ∑ i in t, w i = 1) {p : ι → E} (hp : ∀ i ∈ t, p i ∈ s) :
-  s.linear_combination p w ∈ convex_hull 𝕜 s :=
-(convex_convex_hull 𝕜 s).linear_combination_mem hw₀ hws (λ i hi, subset_convex_hull 𝕜 s $ hp i hi)
+lemma linear_combination_mem_convex_hull (t : finset ι) {w : ι → 𝕜} (hw₀ : ∀ i ∈ t, 0 ≤ w i)
+  (hwt : ∑ i in t, w i = 1) {p : ι → E} (hp : ∀ i ∈ t, p i ∈ s) :
+  t.linear_combination p w ∈ convex_hull 𝕜 s :=
+(convex_convex_hull 𝕜 s).linear_combination_mem hw₀ hwt (λ i hi, subset_convex_hull 𝕜 s $ hp i hi)
 
 -- TODO : Do we need other versions of the next lemma?
 
-/-- Convex hull of `s` is equal to the set of all centers of masses of `finset`s `t`, `p '' t ⊆ s`.
-This version allows finsets in any type in any universe. -/
-lemma convex_hull_eq (s : set E) :
-  convex_hull 𝕜 s = {x : E | ∃ (ι : Type u') (t : finset ι) (w : ι → 𝕜) (p : ι → E)
+/-- Convex hull of `s` is equal to the set of all linear combinations with sum `1` of `finset`s `t`
+where `p '' t ⊆ s`. This version allows finsets in any type in any universe. -/
+lemma _root_.convex_hull_eq (s : set E) :
+  convex_hull 𝕜 s = {x : E | ∃ (ι : Type*) (t : finset ι) (w : ι → 𝕜) (p : ι → E)
     (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hp : ∀ i ∈ t, p i ∈ s),
-    s.linear_combination p w = x} :=
+    t.linear_combination p w = x} :=
 begin
-  refine subset.antisymm (convex_hull_min _ _) _,
+  refine (convex_hull_min _ _).antisymm _,
   { intros x hx,
     use [punit, {punit.star}, λ _, 1, λ _, x, λ _ _, zero_le_one,
       finset.sum_singleton, λ _ _, hx],
     simp only [finset.linear_combination, finset.sum_singleton, inv_one, one_smul] },
   { rintros x y ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ ⟨ι', sy, wy, zy, hwy₀, hwy₁, hzy, rfl⟩
       a b ha hb hab,
-    rw [finset.linear_combination_segment' _ _ _ _ _ _ hwx₁ hwy₁ _ _ hab],
+    rw [finset.linear_combination_segment' _ _ _ _ _ _ _ _ hab],
     refine ⟨_, _, _, _, _, _, _, rfl⟩,
     { rintros i hi,
       rw [finset.mem_union, finset.mem_map, finset.mem_map] at hi,
-      rcases hi with ⟨j, hj, rfl⟩|⟨j, hj, rfl⟩;
-        simp only [sum.elim_inl, sum.elim_inr];
-        apply_rules [mul_nonneg, hwx₀, hwy₀] },
+      obtain ⟨j, hj, rfl⟩ | ⟨j, hj, rfl⟩ := hi,
+      { simp only [sum.elim_inl, function.embedding.inl_apply],
+        exact mul_nonneg ha (hwx₀ j hj) },
+      { simp only [sum.elim_inr, function.embedding.inr_apply],
+        exact mul_nonneg hb (hwy₀ j hj) } },
     { simp [finset.sum_sum_elim, finset.mul_sum.symm, *] },
     { intros i hi,
       rw [finset.mem_union, finset.mem_map, finset.mem_map] at hi,
       rcases hi with ⟨j, hj, rfl⟩|⟨j, hj, rfl⟩; apply_rules [hzx, hzy] } },
-  { rintros _ ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩,
-    exact s.linear_combination_mem_convex_hull hw₀ (hw₁.symm ▸ zero_lt_one) hp }
+  { rintros _ ⟨ι, t, w, z, hw₀, hw₁, hp, rfl⟩,
+    exact t.linear_combination_mem_convex_hull hw₀ hw₁ hp }
 end
 
-lemma finset.convex_hull_eq (s : finset E) :
+protected lemma convex_hull_eq (s : finset E) :
   convex_hull 𝕜 ↑s = {x : E | ∃ (w : E → 𝕜) (hw₀ : ∀ y ∈ s, 0 ≤ w y) (hw₁ : ∑ y in s, w y = 1),
-    s.linear_combination w id = x} :=
+    s.linear_combination id w = x} :=
 begin
-  refine subset.antisymm (convex_hull_min _ _) _,
+  refine (convex_hull_min _ _).antisymm _,
   { intros x hx,
     rw [finset.mem_coe] at hx,
-    refine ⟨_, _, _, finset.linear_combination_ite_eq _ _ _ hx⟩,
+    refine ⟨_, _, _, finset.linear_combination_ite_eq _ _ id hx⟩,
     { intros, split_ifs, exacts [zero_le_one, le_refl 0] },
     { rw [finset.sum_ite_eq, if_pos hx] } },
-  { rintros x y ⟨wx, hwx₀, hwx₁, rfl⟩ ⟨wy, hwy₀, hwy₁, rfl⟩
-      a b ha hb hab,
-    rw [finset.linear_combination_segment _ _ _ _ hwx₁ hwy₁ _ _ hab],
+  { rintro x y ⟨wx, hwx₀, hwx₁, rfl⟩ ⟨wy, hwy₀, hwy₁, rfl⟩ a b ha hb hab,
+    rw [finset.linear_combination_segment _ _ _ _ _ _ hab],
     refine ⟨_, _, _, rfl⟩,
     { rintros i hi,
       apply_rules [add_nonneg, mul_nonneg, hwx₀, hwy₀], },
     { simp only [finset.sum_add_distrib, finset.mul_sum.symm, mul_one, *] } },
   { rintros _ ⟨w, hw₀, hw₁, rfl⟩,
     exact s.linear_combination_mem_convex_hull (λ x hx, hw₀ _ hx)
-      (hw₁.symm ▸ zero_lt_one) (λ x hx, hx) }
+      hw₁ (λ x hx, hx) }
 end
 
-lemma set.finite.convex_hull_eq {s : set E} (hs : finite s) :
+lemma _root_.set.finite.convex_hull_eq {s : set E} (hs : finite s) :
   convex_hull 𝕜 s = {x : E | ∃ (w : E → 𝕜) (hw₀ : ∀ y ∈ s, 0 ≤ w y)
-    (hw₁ : ∑ y in hs.to_finset, w y = 1), hs.to_finset.linear_combination w id = x} :=
+    (hw₁ : ∑ y in hs.to_finset, w y = 1), hs.to_finset.linear_combination id w = x} :=
 by simpa only [set.finite.coe_to_finset, set.finite.mem_to_finset, exists_prop]
   using hs.to_finset.convex_hull_eq
 
 /-- A weak version of Carathéodory's theorem. -/
-lemma convex_hull_eq_union_convex_hull_finite_subsets (s : set E) :
+lemma _root_.convex_hull_eq_union_convex_hull_finite_subsets (s : set E) :
   convex_hull 𝕜 s = ⋃ (t : finset E) (w : ↑t ⊆ s), convex_hull 𝕜 ↑t :=
 begin
-  refine subset.antisymm _ _,
+  refine set.subset.antisymm _ _,
   { rw convex_hull_eq,
-    rintros x ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩,
+    rintros x ⟨ι, t, w, p, hw₀, hw₁, hp, rfl⟩,
     simp only [mem_Union],
-    refine ⟨t.image z, _, _⟩,
+    refine ⟨t.image p, _, _⟩,
     { rw [coe_image, set.image_subset_iff],
       exact hp },
-    { apply s.linear_combination_mem_convex_hull hw₀,
+    { apply t.linear_combination_mem_convex_hull hw₀,
       { simp only [hw₁, zero_lt_one] },
       { exact λ i hi, finset.mem_coe.2 (finset.mem_image_of_mem _ hi) } } },
-   { exact Union_subset (λ i, Union_subset convex_hull_mono), },
+   { exact Union_subset (λ i, Union_subset convex_hull_mono) }
 end
 
+end linear_ordered_field
+end finset
+
 /-! ### `std_simplex` -/
 
-variables (ι) [fintype ι] {f : ι → 𝕜}
+section linear_ordered_field
+variables (𝕜 E ι : Type*) [linear_ordered_field 𝕜] [add_comm_monoid E] [module 𝕜 E] [fintype ι]
+  (s : finset ι)
 
 /-- `std_simplex ι` is the convex hull of the canonical basis in `ι → 𝕜`. -/
 lemma convex_hull_basis_eq_std_simplex :
-  convex_hull 𝕜 (range $ λ(i j:ι), if i = j then (1:𝕜) else 0) = std_simplex ι :=
+  convex_hull 𝕜 (range $ λ(i j:ι), if i = j then (1:𝕜) else (0 : 𝕜)) = std_simplex 𝕜 ι :=
 begin
-  refine subset.antisymm (convex_hull_min _ (convex_std_simplex ι)) _,
+  refine (convex_hull_min _ (convex_std_simplex 𝕜 ι)).antisymm _,
   { rintros _ ⟨i, rfl⟩,
-    exact ite_eq_mem_std_simplex i },
+    exact ite_eq_mem_std_simplex 𝕜 i },
   { rintros w ⟨hw₀, hw₁⟩,
-    rw [pi_eq_sum_univ w, ← finset.univ.linear_combination_eq_of_sum_1 _ hw₁],
+    rw [pi_eq_sum_univ w],
     exact finset.univ.linear_combination_mem_convex_hull (λ i hi, hw₀ i)
-      (hw₁.symm ▸ zero_lt_one) (λ i hi, mem_range_self i) }
+      hw₁ (λ i hi, mem_range_self i) }
 end
 
-variable {ι}
+variables {𝕜 E ι}
 
 /-- The convex hull of a finite set is the image of the standard simplex in `s → 𝕜`
 under the linear map sending each function `w` to `∑ x in s, w x • x`.
@@ -282,7 +302,7 @@ The map is defined in terms of operations on `(s → 𝕜) →ₗ[𝕜] 𝕜` so
 to prove that this map is linear. -/
 lemma set.finite.convex_hull_eq_image {s : set E} (hs : finite s) :
   convex_hull 𝕜 s = by haveI := hs.fintype; exact
-    (⇑(∑ x : s, (@linear_map.proj 𝕜 s _ (λ i, 𝕜) _ _ x).smul_right x.1)) '' (std_simplex s) :=
+    (⇑(∑ x : s, (@linear_map.proj 𝕜 s _ (λ i, 𝕜) _ _ x).smul_right x.1)) '' (std_simplex 𝕜 s) :=
 begin
   rw [← convex_hull_basis_eq_std_simplex, ← linear_map.convex_hull_image, ← set.range_comp, (∘)],
   apply congr_arg,
@@ -292,6 +312,8 @@ begin
 end
 
 /-- All values of a function `f ∈ std_simplex ι` belong to `[0, 1]`. -/
-lemma mem_Icc_of_mem_std_simplex (hf : f ∈ std_simplex ι) (x) :
+lemma mem_Icc_of_mem_std_simplex {f : ι → 𝕜} (hf : f ∈ std_simplex 𝕜 ι) (x) :
   f x ∈ Icc (0 : 𝕜) 1 :=
 ⟨hf.1 x, hf.2 ▸ finset.single_le_sum (λ y hy, hf.1 y) (finset.mem_univ x)⟩
+
+end linear_ordered_field
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index 004a86b2c4207..e2d5cd99226cf 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -13,14 +13,14 @@ inequality. The integral version can be found in `analysis.convex.integral`.
 
 A function `f : E → β` is `convex_on` a set `s` if `s` is itself a convex set, and for any two
 points `x y ∈ s`, the segment joining `(x, f x)` to `(y, f y)` is above the graph of `f`.
-Equivalently, `convex_on 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2}` is
+Equivalently, `convex_on 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f z.1 ≤ p.2}` is
 a convex set.
 
 ## Main declarations
 
 * `convex_on 𝕜 s f`: The function `f` is convex on `s` with scalars `𝕜`.
 * `concave_on 𝕜 s f`: The function `f` is concave on `s` with scalars `𝕜`.
-* `convex_on.map_center_mass_le` `convex_on.map_sum_le`: Convex Jensen's inequality.
+* `convex_on.map_linear_combination_le` `convex_on.map_sum_le`: Convex Jensen's inequality.
 -/
 
 open finset linear_map set
@@ -131,7 +131,7 @@ lemma linear_order.concave_on_of_lt {f : E → β} [linear_order E] (hs : convex
 
 /-- For a function `f` defined on a convex subset `D` of `ℝ`, if for any three points `x < y < z`
 the slope of the secant line of `f` on `[x, y]` is less than or equal to the slope
-of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of proving convexity
+of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of zroving convexity
 of a function is used in the proof of convexity of a function with a monotone derivative. -/
 lemma convex_on_real_of_slope_mono_adjacent {s : set ℝ} (hs : convex ℝ s) {f : ℝ → ℝ}
   (hf : ∀ {x y z : ℝ}, x ∈ s → z ∈ s → x < y → y < z →
@@ -534,57 +534,57 @@ by simpa only [add_comm] using hf.translate_right
 
 /-! ### Jensen's inequality -/
 
-variables {i j : ι} {c : ℝ} {t : finset ι} {w : ι → ℝ} {z : ι → E}
+variables {i j : ι} {c : ℝ} {t : finset ι} {w : ι → ℝ} {p : ι → E}
 
-/-- Convex **Jensen's inequality**, `finset.center_mass` version. -/
-lemma convex_on.map_center_mass_le {f : E → ℝ} (hf : convex_on s f)
-  (h₀ : ∀ i ∈ t, 0 ≤ w i) (hpos : 0 < ∑ i in t, w i)
-  (hmem : ∀ i ∈ t, z i ∈ s) : f (t.center_mass w z) ≤ t.center_mass w (f ∘ z) :=
+/-- Convex **Jensen's inequality**, `finset.linear_combination` version. -/
+lemma convex_on.map_linear_combination_le {f : E → ℝ} (hf : convex_on s f)
+  (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
+  (hmem : ∀ i ∈ t, p i ∈ s) : f (t.linear_combination p w) ≤ t.linear_combination (f ∘ p) w :=
 begin
-  have hmem' : ∀ i ∈ t, (z i, (f ∘ z) i) ∈ {p : E × ℝ | p.1 ∈ s ∧ f p.1 ≤ p.2},
+  have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × ℝ | p.1 ∈ s ∧ f p.1 ≤ p.2},
     from λ i hi, ⟨hmem i hi, le_rfl⟩,
-  convert (hf.convex_epigraph.center_mass_mem h₀ hpos hmem').2;
-    simp only [center_mass, function.comp, prod.smul_fst, prod.fst_sum, prod.smul_snd, prod.snd_sum]
+  convert (hf.convex_epigraph.linear_combination_mem h₀ h₁ hmem').2;
+    simp only [linear_combination, function.comp, prod.smul_fst, prod.fst_sum,
+      prod.smul_snd, prod.snd_sum]
 end
 
 /-- Convex **Jensen's inequality**, `finset.sum` version. -/
 lemma convex_on.map_sum_le {f : E → ℝ} (hf : convex_on s f)
   (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
-  (hmem : ∀ i ∈ t, z i ∈ s) : f (∑ i in t, w i • z i) ≤ ∑ i in t, w i * (f (z i)) :=
-by simpa only [center_mass, h₁, inv_one, one_smul]
-  using hf.map_center_mass_le h₀ (h₁.symm ▸ zero_lt_one) hmem
+  (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i in t, w i • p i) ≤ ∑ i in t, w i * (f (p i)) :=
+by simpa only [linear_combination, h₁, inv_one, one_smul]
+  using hf.map_linear_combination_le h₀ h₁ hmem
 
-/-! ### Maximal principle -/
+/-! ### Maximum principle -/
 
 /-- If a function `f` is convex on `s` takes value `y` at the center of mass of some points
-`z i ∈ s`, then for some `i` we have `y ≤ f (z i)`. -/
-lemma convex_on.exists_ge_of_center_mass {f : E → ℝ} (h : convex_on s f)
-  (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hws : 0 < ∑ i in t, w i) (hz : ∀ i ∈ t, z i ∈ s) :
-  ∃ i ∈ t, f (t.center_mass w z) ≤ f (z i) :=
+`p i ∈ s`, then for some `i` we have `y ≤ f (p i)`. -/
+lemma convex_on.exists_ge_of_linear_combination {f : E → ℝ} (h : convex_on s f)
+  (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hz : ∀ i ∈ t, p i ∈ s) :
+  ∃ i ∈ t, f (t.linear_combination p w) ≤ f (p i) :=
 begin
-  set y := t.center_mass w z,
-  have : f y ≤ t.center_mass w (f ∘ z) := h.map_center_mass_le hw₀ hws hz,
-  rw ← sum_filter_ne_zero at hws,
-  rw [← finset.center_mass_filter_ne_zero (f ∘ z), center_mass, smul_eq_mul,
-    ← div_eq_inv_mul, le_div_iff hws, mul_sum] at this,
-  replace : ∃ i ∈ t.filter (λ i, w i ≠ 0), f y * w i ≤ w i • (f ∘ z) i :=
-    exists_le_of_sum_le (nonempty_of_sum_ne_zero (ne_of_gt hws)) this,
-  rcases this with ⟨i, hi, H⟩,
-  rw [mem_filter] at hi,
-  use [i, hi.1],
-  simp only [smul_eq_mul, mul_comm (w i)] at H,
-  refine (mul_le_mul_right _).1 H,
-  exact lt_of_le_of_ne (hw₀ i hi.1) hi.2.symm
+  set y := t.linear_combination p w,
+  suffices h : ∃ i ∈ t.filter (λ i, w i ≠ 0), w i • f y ≤ w i • (f ∘ p) i,
+  { obtain ⟨i, hi, hfi⟩ := h,
+    rw mem_filter at hi,
+    exact ⟨i, hi.1, (smul_le_smul_iff_of_pos $ (hw₀ i hi.1).lt_of_ne hi.2.symm).1 hfi⟩ },
+  refine exists_le_of_sum_le _ _,
+  { have := @zero_ne_one ℝ _ _,
+    rw [←hw₁, ←sum_filter_ne_zero] at this,
+    exact nonempty_of_sum_ne_zero this.symm },
+  { rw [←linear_combination, ←linear_combination, finset.linear_combination_filter_ne_zero,
+      finset.linear_combination_filter_ne_zero, linear_combination_const_left, hw₁, one_smul],
+    exact h.map_linear_combination_le hw₀ hw₁ hz }
 end
 
 /-- Maximum principle for convex functions. If a function `f` is convex on the convex hull of `s`,
-then `f` can't have a maximum on `convex_hull s` outside of `s`. -/
+then `f` reaches a maximum on `convex_hull ℝ s` outside of `s`. -/
 lemma convex_on.exists_ge_of_mem_convex_hull {f : E → ℝ} (hf : convex_on (convex_hull ℝ s) f)
   {x} (hx : x ∈ convex_hull ℝ s) : ∃ y ∈ s, f x ≤ f y :=
 begin
-  rw _root_.convex_hull_eq at hx,
-  rcases hx with ⟨α, t, w, z, hw₀, hw₁, hz, rfl⟩,
-  rcases hf.exists_ge_of_center_mass hw₀ (hw₁.symm ▸ zero_lt_one)
-    (λ i hi, subset_convex_hull ℝ s (hz i hi)) with ⟨i, hit, Hi⟩,
-  exact ⟨z i, hz i hit, Hi⟩
+  rw convex_hull_eq at hx,
+  obtain ⟨α, t, w, p, hw₀, hw₁, hp, rfl⟩ := hx,
+  rcases hf.exists_ge_of_linear_combination hw₀ hw₁
+    (λ i hi, subset_convex_hull ℝ s (hp i hi)) with ⟨i, hit, Hi⟩,
+  exact ⟨p i, hp i hit, Hi⟩
 end
diff --git a/src/analysis/convex/independent.lean b/src/analysis/convex/independent.lean
index 5820633ea6177..75fb00a7ca937 100644
--- a/src/analysis/convex/independent.lean
+++ b/src/analysis/convex/independent.lean
@@ -33,7 +33,7 @@ convex independence.
 ## TODO
 
 Prove `affine_independent.convex_independent`. This requires some glue between `affine_combination`
-and `finset.center_mass`.
+and `finset.linear_combination`.
 
 ## Tags
 
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index d7cb6f9bd4611..fd27f37592b1c 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -13,8 +13,8 @@ In this file we prove four theorems:
 
 * `convex.smul_integral_mem`: if `μ` is a non-zero finite measure on `α`, `s` is a convex closed set
   in `E`, and `f` is an integrable function sending `μ`-a.e. points to `s`, then the average value
-  of `f` belongs to `s`: `(μ univ).to_real⁻¹ • ∫ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem`
-  for a finite sum version of this lemma.
+  of `f` belongs to `s`: `(μ univ).to_real⁻¹ • ∫ x, f x ∂μ ∈ s`. See also
+  `convex.linear_combination_mem` for a finite sum version of this lemma.
 
 * `convex.integral_mem`: if `μ` is a probability measure on `α`, `s` is a convex closed set in `E`,
   and `f` is an integrable function sending `μ`-a.e. points to `s`, then the expected value of `f`
@@ -25,7 +25,7 @@ In this file we prove four theorems:
   continuous on a convex closed set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is
   a function sending `μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is
   less than or equal to the average value of `g ∘ f` provided that both `f` and `g ∘ f` are
-  integrable. See also `convex.map_center_mass_le` for a finite sum version of this lemma.
+  integrable. See also `convex.map_linear_combination_le` for a finite sum version of this lemma.
 
 * `convex_on.map_integral_le`: Jensen's inequality: if a function `g : E → ℝ` is convex and
   continuous on a convex closed set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a
@@ -66,7 +66,7 @@ begin
     by rw [← (F n).sum_range_measure_preimage_singleton, @ennreal.to_real_sum _ _
       (λ y, μ ((F n) ⁻¹' {y})) (λ _ _, (measure_lt_top _ _))],
   rw [← this, simple_func.integral],
-  refine hs.center_mass_mem (λ _ _, ennreal.to_real_nonneg) _ _,
+  refine hs.linear_combination_mem (λ _ _, ennreal.to_real_nonneg) _ _,
   { rw [this, ennreal.to_real_pos_iff, pos_iff_ne_zero, ne.def, measure.measure_univ_eq_zero],
     exact ⟨hμ, measure_ne_top _ _⟩ },
   { simp only [simple_func.mem_range],
@@ -76,8 +76,8 @@ end
 
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
-`(μ univ).to_real⁻¹ • ∫ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version
-of this lemma. -/
+`(μ univ).to_real⁻¹ • ∫ x, f x ∂μ ∈ s`. See also `convex.linear_combination_mem` for a finite sum
+version of this lemma. -/
 lemma convex.smul_integral_mem
   [is_finite_measure μ] {s : set E} (hs : convex ℝ s) (hsc : is_closed s)
   (hμ : μ ≠ 0) {f : α → E} (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) :
@@ -104,8 +104,8 @@ by simpa [measure_univ] using hs.smul_integral_mem hsc (is_probability_measure.n
 /-- Jensen's inequality: if a function `g : E → ℝ` is convex and continuous on a convex closed set
 `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending `μ`-a.e. points
 to `s`, then the value of `g` at the average value of `f` is less than or equal to the average value
-of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also `convex.map_center_mass_le`
-for a finite sum version of this lemma. -/
+of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
+`convex.map_linear_combination_le` for a finite sum version of this lemma. -/
 lemma convex_on.map_smul_integral_le [is_finite_measure μ] {s : set E} {g : E → ℝ}
   (hg : convex_on s g) (hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) {f : α → E}
   (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
diff --git a/src/analysis/convex/topology.lean b/src/analysis/convex/topology.lean
index 8b5effcbf0858..46caa3544bcda 100644
--- a/src/analysis/convex/topology.lean
+++ b/src/analysis/convex/topology.lean
@@ -41,9 +41,9 @@ section std_simplex
 
 variables [fintype ι]
 
-/-- Every vector in `std_simplex ι` has `max`-norm at most `1`. -/
+/-- Every vector in `std_simplex ℝ ι` has `max`-norm at most `1`. -/
 lemma std_simplex_subset_closed_ball :
-  std_simplex ι ⊆ metric.closed_ball 0 1 :=
+  std_simplex ℝ ι ⊆ metric.closed_ball 0 1 :=
 begin
   assume f hf,
   rw [metric.mem_closed_ball, dist_zero_right],
@@ -55,18 +55,18 @@ end
 
 variable (ι)
 
-/-- `std_simplex ι` is bounded. -/
-lemma bounded_std_simplex : metric.bounded (std_simplex ι) :=
+/-- `std_simplex ℝ ι` is bounded. -/
+lemma bounded_std_simplex : metric.bounded (std_simplex ℝ ι) :=
 (metric.bounded_iff_subset_ball 0).2 ⟨1, std_simplex_subset_closed_ball⟩
 
-/-- `std_simplex ι` is closed. -/
-lemma is_closed_std_simplex : is_closed (std_simplex ι) :=
-(std_simplex_eq_inter ι).symm ▸ is_closed.inter
+/-- `std_simplex ℝ ι` is closed. -/
+lemma is_closed_std_simplex : is_closed (std_simplex ℝ ι) :=
+(std_simplex_eq_inter ℝ ι).symm ▸ is_closed.inter
   (is_closed_Inter $ λ i, is_closed_le continuous_const (continuous_apply i))
   (is_closed_eq (continuous_finset_sum _ $ λ x _, continuous_apply x) continuous_const)
 
-/-- `std_simplex ι` is compact. -/
-lemma compact_std_simplex : is_compact (std_simplex ι) :=
+/-- `std_simplex ℝ ι` is compact. -/
+lemma compact_std_simplex : is_compact (std_simplex ℝ ι) :=
 metric.compact_iff_closed_bounded.2 ⟨is_closed_std_simplex ι, bounded_std_simplex ι⟩
 
 end std_simplex

From 665aeafec8841364947e04e7029bdf7345f37133 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Tue, 21 Sep 2021 16:31:22 +0200
Subject: [PATCH 04/17] break long line

---
 src/analysis/convex/combination.lean | 8 ++++++--
 src/analysis/convex/function.lean    | 2 +-
 2 files changed, 7 insertions(+), 3 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index b4f9a28adca49..9f4ec42d85748 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -9,8 +9,12 @@ import analysis.convex.basic
 # Linear combinations
 
 This file defines linear combinations of points in a semimodule and proves results about convex
-combinations. Convex combinations in a semimodule can be modelled as linear combinations with
-nonnegative coefficients summing to `1`. In an affine space, they can be modelled through affine combinations.
+combinations.
+
+Convex combinations in a semimodule can be modelled as linear combinations with nonnegative coefficients summing to `1`. In an affine space, they can be modelled through affine combinations.
+
+In a vector space, both coincide but it is still an open question whether we can make the two ways
+to derive `convex_combination` definitionally equal.
 
 ## Main declarations
 
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index e2d5cd99226cf..fb9cab39f5528 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -131,7 +131,7 @@ lemma linear_order.concave_on_of_lt {f : E → β} [linear_order E] (hs : convex
 
 /-- For a function `f` defined on a convex subset `D` of `ℝ`, if for any three points `x < y < z`
 the slope of the secant line of `f` on `[x, y]` is less than or equal to the slope
-of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of zroving convexity
+of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of proving convexity
 of a function is used in the proof of convexity of a function with a monotone derivative. -/
 lemma convex_on_real_of_slope_mono_adjacent {s : set ℝ} (hs : convex ℝ s) {f : ℝ → ℝ}
   (hf : ∀ {x y z : ℝ}, x ∈ s → z ∈ s → x < y → y < z →

From e16df20f7a44f9c8fc8b54f57aa8b6c48e4e9b1e Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Tue, 21 Sep 2021 16:45:36 +0200
Subject: [PATCH 05/17] make convex.sum_mem useful

---
 src/analysis/convex/combination.lean | 18 ++++++++++++++----
 1 file changed, 14 insertions(+), 4 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 9f4ec42d85748..3781e5c506241 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -163,16 +163,26 @@ begin
   refine hs hpi _ (h₀ _ (mem_insert_self _ _)) (sum_nonneg ht₀) h₁,
   refine ht (λ j hj, _) _ (λ j hj, hmem _ (mem_insert_of_mem hj)),
   { dsimp,
-    refine mul_nonneg (inv_nonneg.2 (sum_nonneg ht₀)) (ht₀ j hj) },
+    exact mul_nonneg (inv_nonneg.2 (sum_nonneg ht₀)) (ht₀ j hj) },
   dsimp,
   simp, --nonterminal simp to fix
   rw [←mul_sum, inv_eq_one_div, one_div_mul_cancel hsum_t],
 end
 
-lemma _root_.convex.sum_mem (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
+lemma _root_.convex.sum_mem (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i ≠ 0)
   (hp : ∀ i ∈ t, p i ∈ s) :
-  ∑ i in t, w i • p i ∈ s :=
-hs.linear_combination_mem h₀ h₁ hp
+  (∑ i in t, w i)⁻¹ • ∑ i in t, w i • p i ∈ s :=
+begin
+  rw [linear_combination_normalize h₁, inv_smul_smul' h₁],
+  refine hs.linear_combination_mem _ _ hp,
+  {
+    rintro i hi,
+    refine smul_nonneg (h₀ i hi) _,
+    sorry
+  },
+  { rw [smul_sum, div_self h₁],
+  }
+end
 
 lemma _root_.convex_iff_linear_combination_mem :
   convex 𝕜 s ↔

From 4b2bba427a1809a36da5ee2006303909325f8f5f Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Tue, 21 Sep 2021 20:14:43 +0200
Subject: [PATCH 06/17] restore convex.linear_combination_mem

---
 src/analysis/convex/combination.lean | 21 +++++++--------------
 src/analysis/convex/integral.lean    |  6 +++---
 2 files changed, 10 insertions(+), 17 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 3781e5c506241..1fe29bbec967a 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -164,24 +164,17 @@ begin
   refine ht (λ j hj, _) _ (λ j hj, hmem _ (mem_insert_of_mem hj)),
   { dsimp,
     exact mul_nonneg (inv_nonneg.2 (sum_nonneg ht₀)) (ht₀ j hj) },
-  dsimp,
-  simp, --nonterminal simp to fix
-  rw [←mul_sum, inv_eq_one_div, one_div_mul_cancel hsum_t],
+  { simp_rw [pi.smul_apply, ←smul_sum, smul_eq_mul, inv_mul_cancel hsum_t] }
 end
 
-lemma _root_.convex.sum_mem (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i ≠ 0)
-  (hp : ∀ i ∈ t, p i ∈ s) :
-  (∑ i in t, w i)⁻¹ • ∑ i in t, w i • p i ∈ s :=
+lemma _root_.convex.linear_combination_mem' (hs : convex 𝕜 s) (h₀ : ∀ i ∈ t, 0 ≤ w i)
+  (h₁ : ∑ i in t, w i ≠ 0) (hp : ∀ i ∈ t, p i ∈ s) :
+  (∑ i in t, w i)⁻¹ • t.linear_combination p w ∈ s :=
 begin
   rw [linear_combination_normalize h₁, inv_smul_smul' h₁],
-  refine hs.linear_combination_mem _ _ hp,
-  {
-    rintro i hi,
-    refine smul_nonneg (h₀ i hi) _,
-    sorry
-  },
-  { rw [smul_sum, div_self h₁],
-  }
+  refine hs.linear_combination_mem (λ i hi, mul_nonneg (inv_nonneg.2 $ sum_nonneg h₀)
+    (h₀ i hi)) _ hp,
+  simp_rw [pi.smul_apply, ←smul_sum, smul_eq_mul, inv_mul_cancel h₁]
 end
 
 lemma _root_.convex_iff_linear_combination_mem :
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index b35a025ac942e..66ec3aa9c0615 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -18,8 +18,8 @@ In this file we prove four theorems:
 
 * `convex.integral_mem`: if `μ` is a probability measure on `α`, `s` is a convex closed set in `E`,
   and `f` is an integrable function sending `μ`-a.e. points to `s`, then the expected value of `f`
-  belongs to `s`: `∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this
-  lemma.
+  belongs to `s`: `∫ x, f x ∂μ ∈ s`. See also `convex.linear_combination_mem` for a finite sum
+  version of this lemma.
 
 * `convex_on.map_smul_integral_le`: Jensen's inequality: if a function `g : E → ℝ` is convex and
   continuous on a convex closed set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is
@@ -66,7 +66,7 @@ begin
     by rw [← (F n).sum_range_measure_preimage_singleton, @ennreal.to_real_sum _ _
       (λ y, μ ((F n) ⁻¹' {y})) (λ _ _, (measure_ne_top _ _))],
   rw [← this, simple_func.integral],
-  refine hs.linear_combination_mem (λ _ _, ennreal.to_real_nonneg) _ _,
+  refine hs.linear_combination_mem' (λ _ _, ennreal.to_real_nonneg) _ _,
   { rw [this, ennreal.to_real_pos_iff, pos_iff_ne_zero, ne.def, measure.measure_univ_eq_zero],
     exact ⟨hμ, measure_ne_top _ _⟩ },
   { simp only [simple_func.mem_range],

From 56743d978391ab8bd604cf4951eabacb5e5c5e81 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Wed, 22 Sep 2021 09:34:14 +0200
Subject: [PATCH 07/17] fix analysis.convex.integral

---
 src/analysis/convex/combination.lean | 3 ++-
 src/analysis/convex/integral.lean    | 4 ++--
 2 files changed, 4 insertions(+), 3 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 1fe29bbec967a..708d3f31c1461 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -11,7 +11,8 @@ import analysis.convex.basic
 This file defines linear combinations of points in a semimodule and proves results about convex
 combinations.
 
-Convex combinations in a semimodule can be modelled as linear combinations with nonnegative coefficients summing to `1`. In an affine space, they can be modelled through affine combinations.
+Convex combinations in a semimodule can be modelled as linear combinations with nonnegative
+coefficients summing to `1`. In an affine space, they can be modelled through affine combinations.
 
 In a vector space, both coincide but it is still an open question whether we can make the two ways
 to derive `convex_combination` definitionally equal.
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index 66ec3aa9c0615..793bbe5e9e42f 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -67,8 +67,8 @@ begin
       (λ y, μ ((F n) ⁻¹' {y})) (λ _ _, (measure_ne_top _ _))],
   rw [← this, simple_func.integral],
   refine hs.linear_combination_mem' (λ _ _, ennreal.to_real_nonneg) _ _,
-  { rw [this, ennreal.to_real_pos_iff, pos_iff_ne_zero, ne.def, measure.measure_univ_eq_zero],
-    exact ⟨hμ, measure_ne_top _ _⟩ },
+  { rw [this, ne.def, ennreal.to_real_eq_zero_iff, measure.measure_univ_eq_zero],
+    exact not_or hμ (measure_ne_top _ _) },
   { simp only [simple_func.mem_range],
     rintros _ ⟨x, rfl⟩,
     exact simple_func.approx_on_mem hfm h₀ n x }

From 5d3c959cc46da48a983ad6bfd3fac4879ca56e0e Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Wed, 22 Sep 2021 14:11:28 +0200
Subject: [PATCH 08/17] remove unused arguments

---
 src/analysis/convex/combination.lean | 14 +++++---------
 1 file changed, 5 insertions(+), 9 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 708d3f31c1461..c62ef2c991974 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -44,9 +44,6 @@ section ordered_semiring
 variables {𝕜 E ι ι' : Type*} [ordered_semiring 𝕜] [add_comm_monoid E] [module 𝕜 E]
   (i j : ι) (c : 𝕜) (s : finset ι) (p : ι → E) (w : ι → 𝕜)
 
-lemma linear_combination_def :
-  s.linear_combination p w = ∑ i in s, w i • p i := rfl
-
 lemma linear_combination_empty : (∅ : finset ι).linear_combination p w = 0 :=
 by simp only [linear_combination, sum_empty, smul_zero]
 
@@ -71,7 +68,7 @@ by rw [linear_combination, finset.sum_smul]
 /-- A convex combination of two centers of mass is a center of mass as well. This version
 deals with two different index types. -/
 lemma linear_combination_segment' (s : finset ι) (t : finset ι') (ws : ι → 𝕜) (ps : ι → E)
-  (wt : ι' → 𝕜) (pt : ι' → E) (a b : 𝕜) (hab : a + b = 1) :
+  (wt : ι' → 𝕜) (pt : ι' → E) (a b : 𝕜) :
   a • s.linear_combination ps ws + b • t.linear_combination pt wt =
     (s.map function.embedding.inl ∪ t.map function.embedding.inr).linear_combination
       (sum.elim ps pt)
@@ -84,13 +81,12 @@ end
 
 /-- A convex combination of two centers of mass is a center of mass as well. This version
 works if two centers of mass share the set of original points. -/
-lemma linear_combination_segment (s : finset ι) (w₁ w₂ : ι → 𝕜) (p : ι → E) (a b : 𝕜)
-  (hab : a + b = 1) :
+lemma linear_combination_segment (s : finset ι) (w₁ w₂ : ι → 𝕜) (p : ι → E) (a b : 𝕜) :
   a • s.linear_combination p w₁ + b • s.linear_combination p w₂ =
     s.linear_combination p (λ i, a * w₁ i + b * w₂ i) :=
 begin
   unfold linear_combination,
-  simp only [linear_combination_def, smul_sum, sum_add_distrib, add_smul, mul_smul, *],
+  simp only [linear_combination, smul_sum, sum_add_distrib, add_smul, mul_smul, *],
 end
 
 lemma linear_combination_ite_eq (hi : i ∈ s) :
@@ -217,7 +213,7 @@ begin
     simp only [finset.linear_combination, finset.sum_singleton, inv_one, one_smul] },
   { rintros x y ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ ⟨ι', sy, wy, zy, hwy₀, hwy₁, hzy, rfl⟩
       a b ha hb hab,
-    rw [finset.linear_combination_segment' _ _ _ _ _ _ _ _ hab],
+    rw [finset.linear_combination_segment' _ _ _ _ _ _ _ _],
     refine ⟨_, _, _, _, _, _, _, rfl⟩,
     { rintros i hi,
       rw [finset.mem_union, finset.mem_map, finset.mem_map] at hi,
@@ -245,7 +241,7 @@ begin
     { intros, split_ifs, exacts [zero_le_one, le_refl 0] },
     { rw [finset.sum_ite_eq, if_pos hx] } },
   { rintro x y ⟨wx, hwx₀, hwx₁, rfl⟩ ⟨wy, hwy₀, hwy₁, rfl⟩ a b ha hb hab,
-    rw [finset.linear_combination_segment _ _ _ _ _ _ hab],
+    rw [finset.linear_combination_segment _ _ _ _ _ _],
     refine ⟨_, _, _, rfl⟩,
     { rintros i hi,
       apply_rules [add_nonneg, mul_nonneg, hwx₀, hwy₀], },

From 3a4e7b5e68c26dbcffcbea76e1cfed0376a8d6d3 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Wed, 22 Sep 2021 17:55:53 +0200
Subject: [PATCH 09/17] initial commit

---
 src/analysis/convex/basic.lean    |   8 +-
 src/analysis/convex/function.lean | 371 ++++++++++++++++--------------
 2 files changed, 208 insertions(+), 171 deletions(-)

diff --git a/src/analysis/convex/basic.lean b/src/analysis/convex/basic.lean
index 6eca9c847b8e0..7754a4fedd2e0 100644
--- a/src/analysis/convex/basic.lean
+++ b/src/analysis/convex/basic.lean
@@ -119,7 +119,7 @@ begin
   exact h (mem_open_segment_of_ne_left_right 𝕜 hxz hyz hz),
 end
 
-lemma convex.combo_self {x y : 𝕜} (h : x + y = 1) (a : 𝕜) : x • a + y • a = a :=
+lemma convex.combo_self {x y : 𝕜} (h : x + y = 1) (a : E) : x • a + y • a = a :=
 by rw [←add_smul, h, one_smul]
 
 end ordered_semiring
@@ -401,14 +401,14 @@ section ordered_semiring
 variables [ordered_semiring 𝕜]
 
 section add_comm_monoid
-variables [add_comm_monoid E] [module 𝕜 E] [add_comm_monoid F] [module 𝕜 F]
+variables [add_comm_monoid E]
 
 /-- Convexity of sets. -/
-def convex (s : set E) :=
+def convex [has_scalar 𝕜 E](s : set E) :=
 ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
   a • x + b • y ∈ s
 
-variables {𝕜} {s : set E}
+variables {𝕜} [module 𝕜 E] [add_comm_monoid F] [module 𝕜 F] {s : set E}
 
 lemma convex_iff_forall_pos :
   convex 𝕜 s ↔ ∀ ⦃x y⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index fb9cab39f5528..e34b897015d0e 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -13,74 +13,141 @@ inequality. The integral version can be found in `analysis.convex.integral`.
 
 A function `f : E → β` is `convex_on` a set `s` if `s` is itself a convex set, and for any two
 points `x y ∈ s`, the segment joining `(x, f x)` to `(y, f y)` is above the graph of `f`.
-Equivalently, `convex_on 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f z.1 ≤ p.2}` is
+Equivalently, `convex_on 𝕜 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f z.1 ≤ p.2}` is
 a convex set.
 
 ## Main declarations
 
-* `convex_on 𝕜 s f`: The function `f` is convex on `s` with scalars `𝕜`.
-* `concave_on 𝕜 s f`: The function `f` is concave on `s` with scalars `𝕜`.
+* `convex_on 𝕜 𝕜 s f`: The function `f` is convex on `s` with scalars `𝕜`.
+* `concave_on 𝕜 𝕜 s f`: The function `f` is concave on `s` with scalars `𝕜`.
 * `convex_on.map_linear_combination_le` `convex_on.map_sum_le`: Convex Jensen's inequality.
 -/
 
 open finset linear_map set
 open_locale big_operators classical convex pointwise
 
-variables {𝕜 E F ι ι' β : Type*} [add_comm_group E] [module ℝ E] [add_comm_group F] [module ℝ F]
-  [ordered_add_comm_monoid β] [module ℝ β] {s : set E}
+variables (𝕜 : Type*) {E F ι ι' β : Type*}
+
+section ordered_semiring
+variables [ordered_semiring 𝕜] [add_comm_monoid E]
+
+section ordered_add_comm_monoid
+variables [ordered_add_comm_monoid β]
 
 /-- Convexity of functions -/
-def convex_on (s : set E) (f : E → β) : Prop :=
-  convex ℝ s ∧
-  ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
+def convex_on [has_scalar 𝕜 E] [has_scalar 𝕜 β] (s : set E) (f : E → β) : Prop :=
+convex 𝕜 s ∧
+  ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
     f (a • x + b • y) ≤ a • f x + b • f y
 
 /-- Concavity of functions -/
-def concave_on (s : set E) (f : E → β) : Prop :=
-  convex ℝ s ∧
-  ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
+def concave_on [has_scalar 𝕜 E] [has_scalar 𝕜 β] (s : set E) (f : E → β) : Prop :=
+convex 𝕜 s ∧
+  ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
     a • f x + b • f y ≤ f (a • x + b • y)
 
+section has_scalar
+variables {𝕜} [has_scalar 𝕜 E] [has_scalar 𝕜 β] {s : set E}
+
+lemma convex_on_id {s : set 𝕜} (hs : convex 𝕜 s) : convex_on 𝕜 s id := ⟨hs, by { intros, refl }⟩
+
+lemma concave_on_id {s : set 𝕜} (hs : convex 𝕜 s) : concave_on 𝕜 s id := ⟨hs, by { intros, refl }⟩
+
+end has_scalar
+
+section module
+variables {𝕜} [has_scalar 𝕜 E] [module 𝕜 β] {s : set E}
+lemma convex_on_const (c : β) (hs : convex 𝕜 s) : convex_on 𝕜 s (λ x:E, c) :=
+⟨hs, λ x y _ _ a b _ _ hab, (convex.combo_self hab c).ge⟩
+
+lemma concave_on_const (c : β) (hs : convex 𝕜 s) : concave_on 𝕜 s (λ x:E, c) :=
+@convex_on_const _ _ (order_dual β) _ _ _ _ _ _ c hs
+
+end module
+
+section module
+variables {𝕜} [module 𝕜 E] [module 𝕜 β] [linear_order E] {s : set E} {f : E → β}
+
+/-- For a function on a convex set in a linear ordered space, in order to prove that it is convex
+it suffices to verify the inequality `f (a • x + b • y) ≤ a • f x + b • f y` only for `x < y`
+and positive `a`, `b`. The main use case is `E = 𝕜` however one can apply it, e.g., to `𝕜^n` with
+lexicographic order. -/
+lemma linear_order.convex_on_of_lt (hs : convex 𝕜 s)
+  (hf : ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
+    f (a • x + b • y) ≤ a • f x + b • f y) : convex_on 𝕜 s f :=
+begin
+  use hs,
+  intros x y hx hy a b ha hb hab,
+  wlog hxy : x ≤ y using [x y a b, y x b a],
+  { exact le_total _ _ },
+  obtain rfl | hxy := hxy.eq_or_lt,
+  { rw [convex.combo_self hab, convex.combo_self hab] },
+  obtain rfl | ha' := ha.eq_or_lt,
+  { rw [zero_add] at hab, subst b, simp_rw [zero_smul, zero_add, one_smul] },
+  obtain rfl | hb' := hb.eq_or_lt,
+  { rw [add_zero] at hab, subst a },
+  exact hf hx hy hxy ha' hb' hab,
+end
+
+/-- For a function on a convex set in a linear ordered space, in order to prove that it is concave
+it suffices to verify the inequality `a • f x + b • f y ≤ f (a • x + b • y)` only for `x < y`
+and positive `a`, `b`. The main use case is `E = 𝕜` however one can apply it, e.g., to `𝕜^n` with
+lexicographic order. -/
+lemma linear_order.concave_on_of_lt (hs : convex 𝕜 s)
+  (hf : ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
+     a • f x + b • f y ≤ f (a • x + b • y)) : concave_on 𝕜 s f :=
+@linear_order.convex_on_of_lt _ _ (order_dual β) _ _ _ _ _ f _ hs hf
+
+end module
+
+end ordered_add_comm_monoid
+
+section ordered_add_comm_group
+variables [ordered_add_comm_group β] [has_scalar 𝕜 E] [module 𝕜 β]
+
 section
-variables [ordered_smul ℝ β]
+variables {𝕜}
 
-/-- A function `f` is concave iff `-f` is convex. -/
-@[simp] lemma neg_convex_on_iff {γ : Type*} [ordered_add_comm_group γ] [module ℝ γ]
-  (s : set E) (f : E → γ) : convex_on s (-f) ↔ concave_on s f :=
+/-- A function `-f` is convex iff `f` is concave. -/
+@[simp] lemma neg_concave_on_iff
+  (s : set E) (f : E → β) : convex_on 𝕜 s (-f) ↔ concave_on 𝕜 s f :=
 begin
   split,
-  { rintros ⟨hconv, h⟩,
+  { rintro ⟨hconv, h⟩,
     refine ⟨hconv, _⟩,
     intros x y xs ys a b ha hb hab,
     specialize h xs ys ha hb hab,
     simp [neg_apply, neg_le, add_comm] at h,
     exact h },
-  { rintros ⟨hconv, h⟩,
+  { rintro ⟨hconv, h⟩,
     refine ⟨hconv, _⟩,
     intros x y xs ys a b ha hb hab,
     specialize h xs ys ha hb hab,
     simp [neg_apply, neg_le, add_comm, h] }
 end
 
-/-- A function `f` is concave iff `-f` is convex. -/
-@[simp] lemma neg_concave_on_iff {γ : Type*} [ordered_add_comm_group γ] [module ℝ γ]
-  (s : set E) (f : E → γ) : concave_on s (-f) ↔ convex_on s f:=
-by rw [← neg_convex_on_iff s (-f), neg_neg f]
+/-- A function `-f` is concave iff `f` is convex. -/
+@[simp] lemma neg_convex_on_iff {γ : Type*} [ordered_add_comm_group γ] [module 𝕜 γ]
+  (s : set E) (f : E → γ) : concave_on 𝕜 s (-f) ↔ convex_on 𝕜 s f:=
+by rw [← neg_concave_on_iff s (-f), neg_neg f]
 
 end
 
-lemma convex_on_id {s : set ℝ} (hs : convex ℝ s) : convex_on s id := ⟨hs, by { intros, refl }⟩
+end ordered_add_comm_group
+
+end ordered_semiring
 
-lemma concave_on_id {s : set ℝ} (hs : convex ℝ s) : concave_on s id := ⟨hs, by { intros, refl }⟩
+section linear_ordered_field
+variables {𝕜} [linear_ordered_field 𝕜] [add_comm_monoid E]
 
-lemma convex_on_const (c : β) (hs : convex ℝ s) : convex_on s (λ x:E, c) :=
-⟨hs, by { intros, simp only [← add_smul, *, one_smul] }⟩
+section ordered_add_comm_monoid
+variables [ordered_add_comm_monoid β]
 
-lemma concave_on_const (c : β) (hs : convex ℝ s) : concave_on s (λ x:E, c) :=
-@convex_on_const _ (order_dual β) _ _ _ _ _ c hs
+section has_scalar
+variables [has_scalar 𝕜 E] [has_scalar 𝕜 β] {s : set E}
 
 lemma convex_on_iff_div {f : E → β} :
-  convex_on s f ↔ convex ℝ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
+  convex_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
   → f ((a/(a+b)) • x + (b/(a+b)) • y) ≤ (a/(a+b)) • f x + (b/(a+b)) • f y :=
 and_congr iff.rfl
 ⟨begin
@@ -95,48 +162,18 @@ begin
 end⟩
 
 lemma concave_on_iff_div {f : E → β} :
-  concave_on s f ↔ convex ℝ s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
+  concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
   → (a/(a+b)) • f x + (b/(a+b)) • f y ≤ f ((a/(a+b)) • x + (b/(a+b)) • y) :=
-@convex_on_iff_div _ (order_dual β) _ _ _ _ _ _
-
-/-- For a function on a convex set in a linear ordered space, in order to prove that it is convex
-it suffices to verify the inequality `f (a • x + b • y) ≤ a • f x + b • f y` only for `x < y`
-and positive `a`, `b`. The main use case is `E = ℝ` however one can apply it, e.g., to `ℝ^n` with
-lexicographic order. -/
-lemma linear_order.convex_on_of_lt {f : E → β} [linear_order E] (hs : convex ℝ s)
-  (hf : ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → x < y → ∀ ⦃a b : ℝ⦄, 0 < a → 0 < b → a + b = 1 →
-    f (a • x + b • y) ≤ a • f x + b • f y) : convex_on s f :=
-begin
-  use hs,
-  intros x y hx hy a b ha hb hab,
-  wlog hxy : x<=y using [x y a b, y x b a],
-  { exact le_total _ _ },
-  { cases eq_or_lt_of_le hxy with hxy hxy,
-      by { subst y, rw [← add_smul, ← add_smul, hab, one_smul, one_smul] },
-    cases eq_or_lt_of_le ha with ha ha,
-      by { subst a, rw [zero_add] at hab, subst b, simp },
-    cases eq_or_lt_of_le hb with hb hb,
-      by { subst b, rw [add_zero] at hab, subst a, simp },
-    exact hf hx hy hxy ha hb hab }
-end
-
-/-- For a function on a convex set in a linear ordered space, in order to prove that it is concave
-it suffices to verify the inequality `a • f x + b • f y ≤ f (a • x + b • y)` only for `x < y`
-and positive `a`, `b`. The main use case is `E = ℝ` however one can apply it, e.g., to `ℝ^n` with
-lexicographic order. -/
-lemma linear_order.concave_on_of_lt {f : E → β} [linear_order E] (hs : convex ℝ s)
-  (hf : ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → x < y → ∀ ⦃a b : ℝ⦄, 0 < a → 0 < b → a + b = 1 →
-     a • f x + b • f y ≤ f (a • x + b • y)) : concave_on s f :=
-@linear_order.convex_on_of_lt _ (order_dual β) _ _ _ _ _ f _ hs hf
+@convex_on_iff_div _ _ (order_dual β) _ _ _ _ _ _ _
 
-/-- For a function `f` defined on a convex subset `D` of `ℝ`, if for any three points `x < y < z`
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
 the slope of the secant line of `f` on `[x, y]` is less than or equal to the slope
 of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of proving convexity
 of a function is used in the proof of convexity of a function with a monotone derivative. -/
-lemma convex_on_real_of_slope_mono_adjacent {s : set ℝ} (hs : convex ℝ s) {f : ℝ → ℝ}
-  (hf : ∀ {x y z : ℝ}, x ∈ s → z ∈ s → x < y → y < z →
+lemma convex_on_real_of_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜}
+  (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
     (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :
-  convex_on s f :=
+  convex_on 𝕜 s f :=
 linear_order.convex_on_of_lt hs
 begin
   assume x z hx hz hxz a b ha hb hab,
@@ -158,11 +195,11 @@ begin
   convert this; symmetry; simp only [div_eq_iff (ne_of_gt B), y]; ring
 end
 
-/-- For a function `f` defined on a subset `D` of `ℝ`, if `f` is convex on `D`, then for any three
+/-- For a function `f` defined on a subset `D` of `𝕜`, if `f` is convex on `D`, then for any three
 points `x < y < z`, the slope of the secant line of `f` on `[x, y]` is less than or equal to the
 slope of the secant line of `f` on `[x, z]`. -/
-lemma convex_on.slope_mono_adjacent {s : set ℝ} {f : ℝ → ℝ} (hf : convex_on s f)
-  {x y z : ℝ} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
+lemma convex_on.slope_mono_adjacent {s : set 𝕜} {f : 𝕜 → 𝕜} (hf : convex_on 𝕜 s f)
+  {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
   (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) :=
 begin
   have h₁ : 0 < y - x := by linarith,
@@ -186,21 +223,21 @@ begin
   { nlinarith, },
 end
 
-/-- For a function `f` defined on a convex subset `D` of `ℝ`, `f` is convex on `D` iff, for any
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, `f` is convex on `D` iff, for any
 three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is less than or equal to
 the slope,of the secant line of `f` on `[x, z]`. -/
-lemma convex_on_real_iff_slope_mono_adjacent {s : set ℝ} (hs : convex ℝ s) {f : ℝ → ℝ} :
-  convex_on s f ↔
-  (∀ {x y z : ℝ}, x ∈ s → z ∈ s → x < y → y < z →
+lemma convex_on_real_iff_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜} :
+  convex_on 𝕜 s f ↔
+  (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
     (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :=
 ⟨convex_on.slope_mono_adjacent, convex_on_real_of_slope_mono_adjacent hs⟩
 
-/-- For a function `f` defined on a convex subset `D` of `ℝ`, if for any three points `x < y < z`
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
 the slope of the secant line of `f` on `[x, y]` is greater than or equal to the slope
 of the secant line of `f` on `[x, z]`, then `f` is concave on `D`. -/
-lemma concave_on_real_of_slope_mono_adjacent {s : set ℝ} (hs : convex ℝ s) {f : ℝ → ℝ}
-  (hf : ∀ {x y z : ℝ}, x ∈ s → z ∈ s → x < y → y < z →
-    (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : concave_on s f :=
+lemma concave_on_real_of_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜}
+  (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
+    (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : concave_on 𝕜 s f :=
 begin
   rw [←neg_convex_on_iff],
   apply convex_on_real_of_slope_mono_adjacent hs,
@@ -209,11 +246,11 @@ begin
   simp only [hf xs zs xy yz, neg_sub_neg, pi.neg_apply],
 end
 
-/-- For a function `f` defined on a subset `D` of `ℝ`, if `f` is concave on `D`, then for any three
+/-- For a function `f` defined on a subset `D` of `𝕜`, if `f` is concave on `D`, then for any three
 points `x < y < z`, the slope of the secant line of `f` on `[x, y]` is greater than or equal to the
 slope of the secant line of `f` on `[x, z]`. -/
-lemma concave_on.slope_mono_adjacent {s : set ℝ} {f : ℝ → ℝ} (hf : concave_on s f)
-  {x y z : ℝ} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
+lemma concave_on.slope_mono_adjacent {s : set 𝕜} {f : 𝕜 → 𝕜} (hf : concave_on 𝕜 s f)
+  {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
   (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) :=
 begin
   rw [←neg_le_neg_iff, ←neg_div, ←neg_div, neg_sub, neg_sub],
@@ -223,29 +260,29 @@ begin
   apply convex_on.slope_mono_adjacent hf; assumption,
 end
 
-/-- For a function `f` defined on a convex subset `D` of `ℝ`, `f` is concave on `D` iff for any
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, `f` is concave on `D` iff for any
 three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is greater than or equal to
 the slope of the secant line of `f` on `[x, z]`. -/
-lemma concave_on_real_iff_slope_mono_adjacent {s : set ℝ} (hs : convex ℝ s) {f : ℝ → ℝ} :
-  concave_on s f ↔
-  (∀ {x y z : ℝ}, x ∈ s → z ∈ s → x < y → y < z →
+lemma concave_on_real_iff_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜} :
+  concave_on 𝕜 s f ↔
+  (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
     (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) :=
 ⟨concave_on.slope_mono_adjacent, concave_on_real_of_slope_mono_adjacent hs⟩
 
-lemma convex_on.subset {f : E → β} {t : set E} (h_convex_on : convex_on t f)
-  (h_subset : s ⊆ t) (h_convex : convex ℝ s) : convex_on s f :=
+lemma convex_on.subset {f : E → β} {t : set E} (h_convex_on 𝕜 : convex_on 𝕜 t f)
+  (h_subset : s ⊆ t) (h_convex : convex 𝕜 s) : convex_on 𝕜 s f :=
 begin
   apply and.intro h_convex,
   intros x y hx hy,
   exact h_convex_on.2 (h_subset hx) (h_subset hy),
 end
 
-lemma concave_on.subset {f : E → β} {t : set E} (h_concave_on : concave_on t f)
-  (h_subset : s ⊆ t) (h_convex : convex ℝ s) : concave_on s f :=
-@convex_on.subset _ (order_dual β) _ _ _ _ _ f t h_concave_on h_subset h_convex
+lemma concave_on.subset {f : E → β} {t : set E} (h_concave_on 𝕜 : concave_on 𝕜 t f)
+  (h_subset : s ⊆ t) (h_convex : convex 𝕜 s) : concave_on 𝕜 s f :=
+@convex_on.subset _ (order_dual β) _ _ _ _ _ f t h_concave_on 𝕜 h_subset h_convex
 
-lemma convex_on.add {f g : E → β} (hf : convex_on s f) (hg : convex_on s g) :
-  convex_on s (λx, f x + g x) :=
+lemma convex_on.add {f g : E → β} (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
+  convex_on 𝕜 s (λx, f x + g x) :=
 begin
   apply and.intro hf.1,
   intros x y hx hy a b ha hb hab,
@@ -256,12 +293,12 @@ begin
     ... = a • (f x + g x) + b • (f y + g y) : by simp [smul_add, add_assoc]
 end
 
-lemma concave_on.add {f g : E → β} (hf : concave_on s f) (hg : concave_on s g) :
-  concave_on s (λx, f x + g x) :=
+lemma concave_on.add {f g : E → β} (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
+  concave_on 𝕜 s (λx, f x + g x) :=
 @convex_on.add _ (order_dual β) _ _ _ _ _ f g hf hg
 
-lemma convex_on.smul [ordered_smul ℝ β] {f : E → β} {c : ℝ} (hc : 0 ≤ c)
-  (hf : convex_on s f) : convex_on s (λx, c • f x) :=
+lemma convex_on.smul [ordered_smul 𝕜 β] {f : E → β} {c : 𝕜} (hc : 0 ≤ c)
+  (hf : convex_on 𝕜 s f) : convex_on 𝕜 s (λx, c • f x) :=
 begin
   apply and.intro hf.1,
   intros x y hx hy a b ha hb hab,
@@ -271,19 +308,19 @@ begin
     ... = a • (c • f x) + b • (c • f y) : by simp only [smul_add, smul_comm c]
 end
 
-lemma concave_on.smul [ordered_smul ℝ β] {f : E → β} {c : ℝ} (hc : 0 ≤ c)
-  (hf : concave_on s f) : concave_on s (λx, c • f x) :=
+lemma concave_on.smul [ordered_smul 𝕜 β] {f : E → β} {c : 𝕜} (hc : 0 ≤ c)
+  (hf : concave_on 𝕜 s f) : concave_on 𝕜 s (λx, c • f x) :=
 @convex_on.smul _ (order_dual β) _ _ _ _ _ _ f c hc hf
 
 section linear_order
 section monoid
 
-variables {γ : Type*} [linear_ordered_add_comm_monoid γ] [module ℝ γ] [ordered_smul ℝ γ]
+variables {γ : Type*} [linear_ordered_add_comm_monoid γ] [module 𝕜 γ] [ordered_smul 𝕜 γ]
   {f g : E → γ}
 
 /-- The pointwise maximum of convex functions is convex. -/
-lemma convex_on.sup (hf : convex_on s f) (hg : convex_on s g) :
-  convex_on s (f ⊔ g) :=
+lemma convex_on.sup (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
+  convex_on 𝕜 s (f ⊔ g) :=
 begin
    refine ⟨hf.left, λ x y hx hy a b ha hb hab, sup_le _ _⟩,
    { calc f (a • x + b • y) ≤ a • f x + b • f y : hf.right hx hy ha hb hab
@@ -297,12 +334,12 @@ begin
 end
 
 /-- The pointwise minimum of concave functions is concave. -/
-lemma concave_on.inf (hf : concave_on s f) (hg : concave_on s g) :
-  concave_on s (f ⊓ g) :=
+lemma concave_on.inf (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
+  concave_on 𝕜 s (f ⊓ g) :=
 @convex_on.sup _ _ _ _ (order_dual γ) _ _ _ _ _ hf hg
 
 /-- A convex function on a segment is upper-bounded by the max of its endpoints. -/
-lemma convex_on.le_on_segment' (hf : convex_on s f) {x y : E} {a b : ℝ}
+lemma convex_on.le_on_segment' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
   f (a • x + b • y) ≤ max (f x) (f y) :=
 calc
@@ -313,30 +350,30 @@ calc
   ... = max (f x) (f y) : by rw [←add_smul, hab, one_smul]
 
 /-- A concave function on a segment is lower-bounded by the min of its endpoints. -/
-lemma concave_on.le_on_segment' (hf : concave_on s f) {x y : E} {a b : ℝ}
+lemma concave_on.le_on_segment' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
   min (f x) (f y) ≤ f (a • x + b • y) :=
 @convex_on.le_on_segment' _ _ _ _ (order_dual γ) _ _ _ f hf x y a b hx hy ha hb hab
 
 /-- A convex function on a segment is upper-bounded by the max of its endpoints. -/
-lemma convex_on.le_on_segment (hf : convex_on s f) {x y z : E}
-  (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ [x -[ℝ] y]) :
+lemma convex_on.le_on_segment (hf : convex_on 𝕜 s f) {x y z : E}
+  (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ [x -[𝕜] y]) :
   f z ≤ max (f x) (f y) :=
 let ⟨a, b, ha, hb, hab, hz⟩ := hz in hz ▸ hf.le_on_segment' hx hy ha hb hab
 
 /-- A concave function on a segment is lower-bounded by the min of its endpoints. -/
-lemma concave_on.le_on_segment {f : E → γ} (hf : concave_on s f) {x y z : E}
-  (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ [x -[ℝ] y]) :
+lemma concave_on.le_on_segment {f : E → γ} (hf : concave_on 𝕜 s f) {x y z : E}
+  (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ [x -[𝕜] y]) :
     min (f x) (f y) ≤ f z :=
 @convex_on.le_on_segment _ _ _ _ (order_dual γ) _ _ _ f hf x y z hx hy hz
 
 end monoid
 
-variables {γ : Type*} [linear_ordered_cancel_add_comm_monoid γ] [module ℝ γ] [ordered_smul ℝ γ]
+variables {γ : Type*} [linear_ordered_cancel_add_comm_monoid γ] [module 𝕜 γ] [ordered_smul 𝕜 γ]
   {f : E → γ}
 
 -- could be shown without contradiction but yeah
-lemma convex_on.le_left_of_right_le' (hf : convex_on s f) {x y : E} {a b : ℝ}
+lemma convex_on.le_left_of_right_le' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
   (hxy : f y ≤ f (a • x + b • y)) :
   f (a • x + b • y) ≤ f x :=
@@ -350,13 +387,13 @@ begin
     ... = f (a • x + b • y) : by rw [←add_smul, hab, one_smul],
 end
 
-lemma concave_on.left_le_of_le_right' (hf : concave_on s f) {x y : E} {a b : ℝ}
+lemma concave_on.left_le_of_le_right' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
   (hxy : f (a • x + b • y) ≤ f y) :
   f x ≤ f (a • x + b • y) :=
 @convex_on.le_left_of_right_le' _ _ _ _ (order_dual γ) _ _ _ f hf x y a b hx hy ha hb hab hxy
 
-lemma convex_on.le_right_of_left_le' (hf : convex_on s f) {x y : E} {a b : ℝ}
+lemma convex_on.le_right_of_left_le' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
   (hxy : f x ≤ f (a • x + b • y)) :
   f (a • x + b • y) ≤ f y :=
@@ -365,42 +402,42 @@ begin
   exact hf.le_left_of_right_le' hy hx hb ha hab hxy,
 end
 
-lemma concave_on.le_right_of_left_le' (hf : concave_on s f) {x y : E} {a b : ℝ}
+lemma concave_on.le_right_of_left_le' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
   (hxy : f (a • x + b • y) ≤ f x) :
   f y ≤ f (a • x + b • y) :=
 @convex_on.le_right_of_left_le' _ _ _ _ (order_dual γ) _ _ _ f hf x y a b hx hy ha hb hab hxy
 
-lemma convex_on.le_left_of_right_le (hf : convex_on s f) {x y z : E} (hx : x ∈ s)
-  (hy : y ∈ s) (hz : z ∈ open_segment ℝ x y) (hyz : f y ≤ f z) :
+lemma convex_on.le_left_of_right_le (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
+  (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f y ≤ f z) :
   f z ≤ f x :=
 begin
   obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz,
   exact hf.le_left_of_right_le' hx hy ha hb.le hab hyz,
 end
 
-lemma concave_on.left_le_of_le_right (hf : concave_on s f) {x y z : E} (hx : x ∈ s)
-  (hy : y ∈ s) (hz : z ∈ open_segment ℝ x y) (hyz : f z ≤ f y) :
+lemma concave_on.left_le_of_le_right (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
+  (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f z ≤ f y) :
   f x ≤ f z :=
 @convex_on.le_left_of_right_le _ _ _ _ (order_dual γ) _ _ _ f hf x y z hx hy hz hyz
 
-lemma convex_on.le_right_of_left_le (hf : convex_on s f) {x y z : E} (hx : x ∈ s)
-  (hy : y ∈ s) (hz : z ∈ open_segment ℝ x y) (hxz : f x ≤ f z) :
+lemma convex_on.le_right_of_left_le (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
+  (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f x ≤ f z) :
   f z ≤ f y :=
 begin
   obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz,
   exact hf.le_right_of_left_le' hx hy ha.le hb hab hxz,
 end
 
-lemma concave_on.le_right_of_left_le (hf : concave_on s f) {x y z : E} (hx : x ∈ s)
-  (hy : y ∈ s) (hz : z ∈ open_segment ℝ x y) (hxz : f z ≤ f x) :
+lemma concave_on.le_right_of_left_le (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
+  (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f z ≤ f x) :
   f y ≤ f z :=
 @convex_on.le_right_of_left_le _ _ _ _ (order_dual γ) _ _ _ f hf x y z hx hy hz hxz
 
 end linear_order
 
-lemma convex_on.convex_le [ordered_smul ℝ β] {f : E → β} (hf : convex_on s f) (r : β) :
-  convex ℝ {x ∈ s | f x ≤ r} :=
+lemma convex_on.convex_le [ordered_smul 𝕜 β] {f : E → β} (hf : convex_on 𝕜 s f) (r : β) :
+  convex 𝕜 {x ∈ s | f x ≤ r} :=
 λ x y hx hy a b ha hb hab,
 begin
   refine ⟨hf.1 hx.1 hy.1 ha hb hab, _⟩,
@@ -411,13 +448,13 @@ begin
                   ... ≤ r                     : by simp [←add_smul, hab]
 end
 
-lemma concave_on.concave_le [ordered_smul ℝ β] {f : E → β} (hf : concave_on s f) (r : β) :
-  convex ℝ {x ∈ s | r ≤ f x} :=
+lemma concave_on.concave_le [ordered_smul 𝕜 β] {f : E → β} (hf : concave_on 𝕜 s f) (r : β) :
+  convex 𝕜 {x ∈ s | r ≤ f x} :=
 @convex_on.convex_le _ (order_dual β) _ _ _ _ _ _ f hf r
 
 lemma convex_on.convex_lt {γ : Type*} [ordered_cancel_add_comm_monoid γ]
-  [module ℝ γ] [ordered_smul ℝ γ]
-  {f : E → γ} (hf : convex_on s f) (r : γ) : convex ℝ {x ∈ s | f x < r} :=
+  [module 𝕜 γ] [ordered_smul 𝕜 γ]
+  {f : E → γ} (hf : convex_on 𝕜 s f) (r : γ) : convex 𝕜 {x ∈ s | f x < r} :=
 begin
   intros a b as bs xa xb hxa hxb hxaxb,
   refine ⟨hf.1 as.1 bs.1 hxa hxb hxaxb, _⟩,
@@ -436,14 +473,14 @@ begin
 end
 
 lemma concave_on.convex_lt {γ : Type*} [ordered_cancel_add_comm_monoid γ]
-  [module ℝ γ] [ordered_smul ℝ γ]
-  {f : E → γ} (hf : concave_on s f) (r : γ) : convex ℝ {x ∈ s | r < f x} :=
+  [module 𝕜 γ] [ordered_smul 𝕜 γ]
+  {f : E → γ} (hf : concave_on 𝕜 s f) (r : γ) : convex 𝕜 {x ∈ s | r < f x} :=
 @convex_on.convex_lt _ _ _ _ (order_dual γ) _ _ _ f hf r
 
 lemma convex_on.convex_epigraph {γ : Type*} [ordered_add_comm_group γ]
-  [module ℝ γ] [ordered_smul ℝ γ]
-  {f : E → γ} (hf : convex_on s f) :
-  convex ℝ {p : E × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
+  [module 𝕜 γ] [ordered_smul 𝕜 γ]
+  {f : E → γ} (hf : convex_on 𝕜 s f) :
+  convex 𝕜 {p : E × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
 begin
   rintros ⟨x, r⟩ ⟨y, t⟩ ⟨hx, hr⟩ ⟨hy, ht⟩ a b ha hb hab,
   refine ⟨hf.1 hx hy ha hb hab, _⟩,
@@ -453,15 +490,15 @@ begin
 end
 
 lemma concave_on.convex_hypograph {γ : Type*} [ordered_add_comm_group γ]
-  [module ℝ γ] [ordered_smul ℝ γ]
-  {f : E → γ} (hf : concave_on s f) :
-  convex ℝ {p : E × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
+  [module 𝕜 γ] [ordered_smul 𝕜 γ]
+  {f : E → γ} (hf : concave_on 𝕜 s f) :
+  convex 𝕜 {p : E × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
 @convex_on.convex_epigraph _ _ _ _ (order_dual γ) _ _ _ f hf
 
 lemma convex_on_iff_convex_epigraph {γ : Type*} [ordered_add_comm_group γ]
-  [module ℝ γ] [ordered_smul ℝ γ]
+  [module 𝕜 γ] [ordered_smul 𝕜 γ]
   {f : E → γ} :
-  convex_on s f ↔ convex ℝ {p : E × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
+  convex_on 𝕜 s f ↔ convex 𝕜 {p : E × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
 begin
   refine ⟨convex_on.convex_epigraph, λ h, ⟨_, _⟩⟩,
   { assume x y hx hy a b ha hb hab,
@@ -471,22 +508,22 @@ begin
 end
 
 lemma concave_on_iff_convex_hypograph {γ : Type*} [ordered_add_comm_group γ]
-  [module ℝ γ] [ordered_smul ℝ γ]
+  [module 𝕜 γ] [ordered_smul 𝕜 γ]
   {f : E → γ} :
-  concave_on s f ↔ convex ℝ {p : E × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
+  concave_on 𝕜 s f ↔ convex 𝕜 {p : E × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
 @convex_on_iff_convex_epigraph _ _ _ _ (order_dual γ) _ _ _ f
 
 /- A linear map is convex. -/
-lemma linear_map.convex_on (f : E →ₗ[ℝ] β) {s : set E} (hs : convex ℝ s) : convex_on s f :=
+lemma linear_map.convex_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : convex_on 𝕜 s f :=
 ⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩
 
 /- A linear map is concave. -/
-lemma linear_map.concave_on (f : E →ₗ[ℝ] β) {s : set E} (hs : convex ℝ s) : concave_on s f :=
+lemma linear_map.concave_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : concave_on 𝕜 s f :=
 ⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩
 
 /-- If a function is convex on `s`, it remains convex when precomposed by an affine map. -/
-lemma convex_on.comp_affine_map {f : F → β} (g : E →ᵃ[ℝ] F) {s : set F}
-  (hf : convex_on s f) : convex_on (g ⁻¹' s) (f ∘ g) :=
+lemma convex_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F}
+  (hf : convex_on 𝕜 s f) : convex_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
 begin
   refine ⟨hf.1.affine_preimage  _,_⟩,
   intros x y xs ys a b ha hb hab,
@@ -498,50 +535,50 @@ begin
 end
 
 /-- If a function is concave on `s`, it remains concave when precomposed by an affine map. -/
-lemma concave_on.comp_affine_map {f : F → β} (g : E →ᵃ[ℝ] F) {s : set F}
-  (hf : concave_on s f) : concave_on (g ⁻¹' s) (f ∘ g) :=
+lemma concave_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F}
+  (hf : concave_on 𝕜 s f) : concave_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
 @convex_on.comp_affine_map _ _ (order_dual β) _ _ _ _ _ _ f g s hf
 
 /-- If `g` is convex on `s`, so is `(g ∘ f)` on `f ⁻¹' s` for a linear `f`. -/
-lemma convex_on.comp_linear_map {g : F → β} {s : set F} (hg : convex_on s g) (f : E →ₗ[ℝ] F) :
-  convex_on (f ⁻¹' s) (g ∘ f) :=
+lemma convex_on.comp_linear_map {g : F → β} {s : set F} (hg : convex_on 𝕜 s g) (f : E →ₗ[𝕜] F) :
+  convex_on 𝕜 (f ⁻¹' s) (g ∘ f) :=
 hg.comp_affine_map f.to_affine_map
 
 /-- If `g` is concave on `s`, so is `(g ∘ f)` on `f ⁻¹' s` for a linear `f`. -/
-lemma concave_on.comp_linear_map {g : F → β} {s : set F} (hg : concave_on s g) (f : E →ₗ[ℝ] F) :
-  concave_on (f ⁻¹' s) (g ∘ f) :=
+lemma concave_on.comp_linear_map {g : F → β} {s : set F} (hg : concave_on 𝕜 s g) (f : E →ₗ[𝕜] F) :
+  concave_on 𝕜 (f ⁻¹' s) (g ∘ f) :=
 hg.comp_affine_map f.to_affine_map
 
 /-- If a function is convex on `s`, it remains convex after a translation. -/
-lemma convex_on.translate_right {f : E → β} {s : set E} {a : E} (hf : convex_on s f) :
-  convex_on ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
-hf.comp_affine_map $ affine_map.const ℝ E a +ᵥ affine_map.id ℝ E
+lemma convex_on.translate_right {f : E → β} {s : set E} {a : E} (hf : convex_on 𝕜 s f) :
+  convex_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
+hf.comp_affine_map $ affine_map.const 𝕜 E a +ᵥ affine_map.id 𝕜 E
 
 /-- If a function is concave on `s`, it remains concave after a translation. -/
-lemma concave_on.translate_right {f : E → β} {s : set E} {a : E} (hf : concave_on s f) :
-  concave_on ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
-hf.comp_affine_map $ affine_map.const ℝ E a +ᵥ affine_map.id ℝ E
+lemma concave_on.translate_right {f : E → β} {s : set E} {a : E} (hf : concave_on 𝕜 s f) :
+  concave_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
+hf.comp_affine_map $ affine_map.const 𝕜 E a +ᵥ affine_map.id 𝕜 E
 
 /-- If a function is convex on `s`, it remains convex after a translation. -/
-lemma convex_on.translate_left {f : E → β} {s : set E} {a : E} (hf : convex_on s f) :
-  convex_on ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
+lemma convex_on.translate_left {f : E → β} {s : set E} {a : E} (hf : convex_on 𝕜 s f) :
+  convex_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
 by simpa only [add_comm] using hf.translate_right
 
 /-- If a function is concave on `s`, it remains concave after a translation. -/
-lemma concave_on.translate_left {f : E → β} {s : set E} {a : E} (hf : concave_on s f) :
-  concave_on ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
+lemma concave_on.translate_left {f : E → β} {s : set E} {a : E} (hf : concave_on 𝕜 s f) :
+  concave_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
 by simpa only [add_comm] using hf.translate_right
 
 /-! ### Jensen's inequality -/
 
-variables {i j : ι} {c : ℝ} {t : finset ι} {w : ι → ℝ} {p : ι → E}
+variables {i j : ι} {c : 𝕜} {t : finset ι} {w : ι → 𝕜} {p : ι → E}
 
 /-- Convex **Jensen's inequality**, `finset.linear_combination` version. -/
-lemma convex_on.map_linear_combination_le {f : E → ℝ} (hf : convex_on s f)
+lemma convex_on.map_linear_combination_le {f : E → 𝕜} (hf : convex_on 𝕜 s f)
   (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
   (hmem : ∀ i ∈ t, p i ∈ s) : f (t.linear_combination p w) ≤ t.linear_combination (f ∘ p) w :=
 begin
-  have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × ℝ | p.1 ∈ s ∧ f p.1 ≤ p.2},
+  have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × 𝕜 | p.1 ∈ s ∧ f p.1 ≤ p.2},
     from λ i hi, ⟨hmem i hi, le_rfl⟩,
   convert (hf.convex_epigraph.linear_combination_mem h₀ h₁ hmem').2;
     simp only [linear_combination, function.comp, prod.smul_fst, prod.fst_sum,
@@ -549,7 +586,7 @@ begin
 end
 
 /-- Convex **Jensen's inequality**, `finset.sum` version. -/
-lemma convex_on.map_sum_le {f : E → ℝ} (hf : convex_on s f)
+lemma convex_on.map_sum_le {f : E → 𝕜} (hf : convex_on 𝕜 s f)
   (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
   (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i in t, w i • p i) ≤ ∑ i in t, w i * (f (p i)) :=
 by simpa only [linear_combination, h₁, inv_one, one_smul]
@@ -559,7 +596,7 @@ by simpa only [linear_combination, h₁, inv_one, one_smul]
 
 /-- If a function `f` is convex on `s` takes value `y` at the center of mass of some points
 `p i ∈ s`, then for some `i` we have `y ≤ f (p i)`. -/
-lemma convex_on.exists_ge_of_linear_combination {f : E → ℝ} (h : convex_on s f)
+lemma convex_on.exists_ge_of_linear_combination {f : E → 𝕜} (h : convex_on 𝕜 s f)
   (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hz : ∀ i ∈ t, p i ∈ s) :
   ∃ i ∈ t, f (t.linear_combination p w) ≤ f (p i) :=
 begin
@@ -569,7 +606,7 @@ begin
     rw mem_filter at hi,
     exact ⟨i, hi.1, (smul_le_smul_iff_of_pos $ (hw₀ i hi.1).lt_of_ne hi.2.symm).1 hfi⟩ },
   refine exists_le_of_sum_le _ _,
-  { have := @zero_ne_one ℝ _ _,
+  { have := @zero_ne_one 𝕜 _ _,
     rw [←hw₁, ←sum_filter_ne_zero] at this,
     exact nonempty_of_sum_ne_zero this.symm },
   { rw [←linear_combination, ←linear_combination, finset.linear_combination_filter_ne_zero,
@@ -578,13 +615,13 @@ begin
 end
 
 /-- Maximum principle for convex functions. If a function `f` is convex on the convex hull of `s`,
-then `f` reaches a maximum on `convex_hull ℝ s` outside of `s`. -/
-lemma convex_on.exists_ge_of_mem_convex_hull {f : E → ℝ} (hf : convex_on (convex_hull ℝ s) f)
-  {x} (hx : x ∈ convex_hull ℝ s) : ∃ y ∈ s, f x ≤ f y :=
+then `f` reaches a maximum on `convex_hull 𝕜 s` outside of `s`. -/
+lemma convex_on.exists_ge_of_mem_convex_hull {f : E → 𝕜} (hf : convex_on 𝕜 (convex_hull 𝕜 s) f)
+  {x} (hx : x ∈ convex_hull 𝕜 s) : ∃ y ∈ s, f x ≤ f y :=
 begin
   rw convex_hull_eq at hx,
   obtain ⟨α, t, w, p, hw₀, hw₁, hp, rfl⟩ := hx,
   rcases hf.exists_ge_of_linear_combination hw₀ hw₁
-    (λ i hi, subset_convex_hull ℝ s (hp i hi)) with ⟨i, hit, Hi⟩,
+    (λ i hi, subset_convex_hull 𝕜 s (hp i hi)) with ⟨i, hit, Hi⟩,
   exact ⟨p i, hp i hit, Hi⟩
 end

From 7f5ccd4205f4716ea397d9452c09f25575f17684 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Thu, 23 Sep 2021 15:59:50 +0200
Subject: [PATCH 10/17] done with the API

---
 src/analysis/calculus/mean_value.lean |   2 +-
 src/analysis/convex/combination.lean  |  13 +-
 src/analysis/convex/function.lean     | 795 ++++++++++++++------------
 src/analysis/convex/integral.lean     |  13 +-
 4 files changed, 445 insertions(+), 378 deletions(-)

diff --git a/src/analysis/calculus/mean_value.lean b/src/analysis/calculus/mean_value.lean
index d3a16f9eaa4e8..dbf910b1fa619 100644
--- a/src/analysis/calculus/mean_value.lean
+++ b/src/analysis/calculus/mean_value.lean
@@ -974,7 +974,7 @@ theorem convex_on_of_deriv_mono {D : set ℝ} (hD : convex ℝ D) {f : ℝ → 
   (hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
   (hf'_mono : ∀ x y ∈ interior D, x ≤ y → deriv f x ≤ deriv f y) :
   convex_on D f :=
-convex_on_real_of_slope_mono_adjacent hD
+convex_on_of_slope_mono_adjacent hD
 begin
   intros x y z hx hz hxy hyz,
   -- First we prove some trivial inclusions
diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index c62ef2c991974..254b95093721e 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -135,10 +135,21 @@ section linear_ordered_field
 variables {𝕜 E ι ι' : Type*} [linear_ordered_field 𝕜] [add_comm_monoid E] [module 𝕜 E]
   {s : set E} {t : finset ι} {p : ι → E} {w : ι → 𝕜}
 
-lemma linear_combination_normalize  (hw : ∑ i in t, w i ≠ 0) :
+lemma linear_combination_normalize (hw : ∑ i in t, w i ≠ 0) :
   t.linear_combination p w = (∑ i in t, w i) • t.linear_combination p ((∑ i in t, w i)⁻¹ • w) :=
 by rw [linear_combination_smul_right, smul_inv_smul' hw]
 
+lemma linear_combination_normalize_weight_sum (hw : ∑ i in t, w i ≠ 0) :
+  ∑ i in t, ((∑ i in t, w i)⁻¹ • w) i = 1 :=
+by { simp_rw pi.smul_apply, rw [←smul_sum, smul_eq_mul, inv_mul_cancel hw] }
+
+lemma linear_combination_normalize_weight_nonneg (hw : ∀ i ∈ t, 0 ≤ w i) (i : ι) (hi : i ∈ t) :
+  (0 : 𝕜) ≤ ((∑ i in t, w i)⁻¹ • w) i :=
+begin
+  rw [pi.smul_apply, smul_eq_mul],
+  exact mul_nonneg (inv_nonneg.2 $ sum_nonneg hw) (hw i hi),
+end
+
 /-- The linear combination of a finite subset of a convex set belongs to the set
 provided that all weights are non-negative, and the total weight is `1`. -/
 lemma _root_.convex.linear_combination_mem (hs : convex 𝕜 s) :
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index e34b897015d0e..fb521e6d13845 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -26,13 +26,13 @@ a convex set.
 open finset linear_map set
 open_locale big_operators classical convex pointwise
 
-variables (𝕜 : Type*) {E F ι ι' β : Type*}
+variables {𝕜 E F β ι : Type*}
 
 section ordered_semiring
-variables [ordered_semiring 𝕜] [add_comm_monoid E]
+variables [ordered_semiring 𝕜] [add_comm_monoid E] [add_comm_monoid F]
 
 section ordered_add_comm_monoid
-variables [ordered_add_comm_monoid β]
+variables (𝕜) [ordered_add_comm_monoid β]
 
 /-- Convexity of functions -/
 def convex_on [has_scalar 𝕜 E] [has_scalar 𝕜 β] (s : set E) (f : E → β) : Prop :=
@@ -46,17 +46,46 @@ convex 𝕜 s ∧
   ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
     a • f x + b • f y ≤ f (a • x + b • y)
 
+variables {𝕜}
+
 section has_scalar
-variables {𝕜} [has_scalar 𝕜 E] [has_scalar 𝕜 β] {s : set E}
+variables [has_scalar 𝕜 E] [has_scalar 𝕜 β] {s : set E}
 
 lemma convex_on_id {s : set 𝕜} (hs : convex 𝕜 s) : convex_on 𝕜 s id := ⟨hs, by { intros, refl }⟩
 
 lemma concave_on_id {s : set 𝕜} (hs : convex 𝕜 s) : concave_on 𝕜 s id := ⟨hs, by { intros, refl }⟩
 
+lemma convex_on.subset {f : E → β} {t : set E} (hf : convex_on 𝕜 t f) (hst : s ⊆ t)
+  (hs : convex 𝕜 s) : convex_on 𝕜 s f :=
+⟨hs, λ x y hx hy, hf.2 (hst hx) (hst hy)⟩
+
+lemma concave_on.subset {f : E → β} {t : set E} (hf : concave_on 𝕜 t f) (hst : s ⊆ t)
+(hs : convex 𝕜 s) : concave_on 𝕜 s f :=
+⟨hs, λ x y hx hy, hf.2 (hst hx) (hst hy)⟩
+
 end has_scalar
 
+section distrib_mul_action
+variables [has_scalar 𝕜 E] [distrib_mul_action 𝕜 β] {s : set E}
+
+lemma convex_on.add {f g : E → β} (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
+  convex_on 𝕜 s (λ x, f x + g x) :=
+⟨hf.1, λ x y hx hy a b ha hb hab,
+  calc
+    f (a • x + b • y) + g (a • x + b • y) ≤ (a • f x + b • f y) + (a • g x + b • g y)
+      : add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)
+    ... = a • f x + a • g x + b • f y + b • g y : by abel
+    ... = a • (f x + g x) + b • (f y + g y) : by simp only [smul_add, add_assoc]⟩
+
+lemma concave_on.add {f g : E → β} (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
+  concave_on 𝕜 s (λ x, f x + g x) :=
+@convex_on.add _ _ (order_dual β) _ _ _ _ _ _ f g hf hg
+
+end distrib_mul_action
+
 section module
-variables {𝕜} [has_scalar 𝕜 E] [module 𝕜 β] {s : set E}
+variables [has_scalar 𝕜 E] [module 𝕜 β] {s : set E}
+
 lemma convex_on_const (c : β) (hs : convex 𝕜 s) : convex_on 𝕜 s (λ x:E, c) :=
 ⟨hs, λ x y _ _ a b _ _ hab, (convex.combo_self hab c).ge⟩
 
@@ -65,19 +94,95 @@ lemma concave_on_const (c : β) (hs : convex 𝕜 s) : concave_on 𝕜 s (λ x:E
 
 end module
 
+section ordered_smul
+variables [has_scalar 𝕜 E] [module 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f : E → β}
+
+lemma convex_on.convex_le (hf : convex_on 𝕜 s f) (r : β) :
+  convex 𝕜 {x ∈ s | f x ≤ r} :=
+λ x y hx hy a b ha hb hab, ⟨hf.1 hx.1 hy.1 ha hb hab,
+  calc
+    f (a • x + b • y) ≤ a • (f x) + b • (f y) : hf.2 hx.1 hy.1 ha hb hab
+                  ... ≤ a • r + b • r         : add_le_add (smul_le_smul_of_nonneg hx.2 ha)
+                                                  (smul_le_smul_of_nonneg hy.2 hb)
+                  ... = r                     : convex.combo_self hab r⟩
+
+lemma concave_on.concave_le (hf : concave_on 𝕜 s f) (r : β) :
+  convex 𝕜 {x ∈ s | r ≤ f x} :=
+@convex_on.convex_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf r
+
+lemma convex_on.convex_epigraph (hf : convex_on 𝕜 s f) :
+  convex 𝕜 {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
+begin
+  rintro ⟨x, r⟩ ⟨y, t⟩ ⟨hx, hr⟩ ⟨hy, ht⟩ a b ha hb hab,
+  refine ⟨hf.1 hx hy ha hb hab, _⟩,
+  calc f (a • x + b • y) ≤ a • f x + b • f y : hf.2 hx hy ha hb hab
+  ... ≤ a • r + b • t : add_le_add (smul_le_smul_of_nonneg hr ha)
+                            (smul_le_smul_of_nonneg ht hb)
+end
+
+lemma concave_on.convex_hypograph (hf : concave_on 𝕜 s f) :
+  convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
+@convex_on.convex_epigraph 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf
+
+lemma convex_on_iff_convex_epigraph :
+  convex_on 𝕜 s f ↔ convex 𝕜 {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
+⟨convex_on.convex_epigraph, λ h,
+  ⟨λ x y hx hy a b ha hb hab, (@h (x, f x) (y, f y) ⟨hx, le_rfl⟩ ⟨hy, le_rfl⟩ a b ha hb hab).1,
+  λ x y hx hy a b ha hb hab, (@h (x, f x) (y, f y) ⟨hx, le_rfl⟩ ⟨hy, le_rfl⟩ a b ha hb hab).2⟩⟩
+
+lemma concave_on_iff_convex_hypograph :
+  concave_on 𝕜 s f ↔ convex 𝕜 {p : E × β | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
+@convex_on_iff_convex_epigraph 𝕜 E (order_dual β) _ _ _ _ _ _ _ f
+
+end ordered_smul
+
 section module
-variables {𝕜} [module 𝕜 E] [module 𝕜 β] [linear_order E] {s : set E} {f : E → β}
+variables [module 𝕜 E] [module 𝕜 β]
 
-/-- For a function on a convex set in a linear ordered space, in order to prove that it is convex
-it suffices to verify the inequality `f (a • x + b • y) ≤ a • f x + b • f y` only for `x < y`
-and positive `a`, `b`. The main use case is `E = 𝕜` however one can apply it, e.g., to `𝕜^n` with
-lexicographic order. -/
+/-- A linear map is convex. -/
+lemma linear_map.convex_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : convex_on 𝕜 s f :=
+⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩
+
+/-- A linear map is concave. -/
+lemma linear_map.concave_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : concave_on 𝕜 s f :=
+⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩
+
+/-- If a function is convex on `s`, it remains convex after a translation. -/
+lemma convex_on.translate_right {f : E → β} {s : set E} {c : E} (hf : convex_on 𝕜 s f) :
+  convex_on 𝕜 ((λ z, c + z) ⁻¹' s) (f ∘ (λ z, c + z)) :=
+⟨hf.1.translate_preimage_right _, λ x y hx hy a b ha hb hab,
+  calc
+    f (c + (a • x + b • y)) = f (a • (c + x) + b • (c + y))
+        : by rw [smul_add, smul_add, add_add_add_comm, convex.combo_self hab]
+    ... ≤ a • f (c + x) + b • f (c + y) : hf.2 hx hy ha hb hab⟩
+
+/-- If a function is concave on `s`, it remains concave after a translation. -/
+lemma concave_on.translate_right {f : E → β} {s : set E} {a : E} (hf : concave_on 𝕜 s f) :
+  concave_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
+@convex_on.translate_right 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ hf
+
+/-- If a function is convex on `s`, it remains convex after a translation. -/
+lemma convex_on.translate_left {f : E → β} {s : set E} {a : E} (hf : convex_on 𝕜 s f) :
+  convex_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
+by simpa only [add_comm] using hf.translate_right
+
+/-- If a function is concave on `s`, it remains concave after a translation. -/
+lemma concave_on.translate_left {f : E → β} {s : set E} {a : E} (hf : concave_on 𝕜 s f) :
+  concave_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
+by simpa only [add_comm] using hf.translate_right
+
+variables [linear_order E] {s : set E} {f : E → β}
+
+/-- For a function on a convex set in a linear ordered space (where the order and the algebraic
+structures aren't necessarily compatible), in order to prove that it is convex, it suffices to
+verify the inequality `f (a • x + b • y) ≤ a • f x + b • f y` only for `x < y` and positive `a`,
+`b`. The main use case is `E = 𝕜` however one can apply it, e.g., to `𝕜^n` with lexicographic order.
+-/
 lemma linear_order.convex_on_of_lt (hs : convex 𝕜 s)
   (hf : ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
     f (a • x + b • y) ≤ a • f x + b • f y) : convex_on 𝕜 s f :=
 begin
-  use hs,
-  intros x y hx hy a b ha hb hab,
+  refine ⟨hs, λ x y hx hy a b ha hb hab, _⟩,
   wlog hxy : x ≤ y using [x y a b, y x b a],
   { exact le_total _ _ },
   obtain rfl | hxy := hxy.eq_or_lt,
@@ -85,238 +190,71 @@ begin
   obtain rfl | ha' := ha.eq_or_lt,
   { rw [zero_add] at hab, subst b, simp_rw [zero_smul, zero_add, one_smul] },
   obtain rfl | hb' := hb.eq_or_lt,
-  { rw [add_zero] at hab, subst a },
+  { rw [add_zero] at hab, subst a, simp_rw [zero_smul, add_zero, one_smul] },
   exact hf hx hy hxy ha' hb' hab,
 end
 
 /-- For a function on a convex set in a linear ordered space, in order to prove that it is concave
 it suffices to verify the inequality `a • f x + b • f y ≤ f (a • x + b • y)` only for `x < y`
-and positive `a`, `b`. The main use case is `E = 𝕜` however one can apply it, e.g., to `𝕜^n` with
+and positive `a`, `b`. The main use case is `E = ℝ` however one can apply it, e.g., to `ℝ^n` with
 lexicographic order. -/
 lemma linear_order.concave_on_of_lt (hs : convex 𝕜 s)
   (hf : ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
      a • f x + b • f y ≤ f (a • x + b • y)) : concave_on 𝕜 s f :=
-@linear_order.convex_on_of_lt _ _ (order_dual β) _ _ _ _ _ f _ hs hf
+@linear_order.convex_on_of_lt _ _ (order_dual β) _ _ _ _ _ _ s f hs hf
 
 end module
 
-end ordered_add_comm_monoid
-
-section ordered_add_comm_group
-variables [ordered_add_comm_group β] [has_scalar 𝕜 E] [module 𝕜 β]
-
-section
-variables {𝕜}
-
-/-- A function `-f` is convex iff `f` is concave. -/
-@[simp] lemma neg_concave_on_iff
-  (s : set E) (f : E → β) : convex_on 𝕜 s (-f) ↔ concave_on 𝕜 s f :=
-begin
-  split,
-  { rintro ⟨hconv, h⟩,
-    refine ⟨hconv, _⟩,
-    intros x y xs ys a b ha hb hab,
-    specialize h xs ys ha hb hab,
-    simp [neg_apply, neg_le, add_comm] at h,
-    exact h },
-  { rintro ⟨hconv, h⟩,
-    refine ⟨hconv, _⟩,
-    intros x y xs ys a b ha hb hab,
-    specialize h xs ys ha hb hab,
-    simp [neg_apply, neg_le, add_comm, h] }
-end
-
-/-- A function `-f` is concave iff `f` is convex. -/
-@[simp] lemma neg_convex_on_iff {γ : Type*} [ordered_add_comm_group γ] [module 𝕜 γ]
-  (s : set E) (f : E → γ) : concave_on 𝕜 s (-f) ↔ convex_on 𝕜 s f:=
-by rw [← neg_concave_on_iff s (-f), neg_neg f]
-
-end
-
-end ordered_add_comm_group
-
-end ordered_semiring
-
-section linear_ordered_field
-variables {𝕜} [linear_ordered_field 𝕜] [add_comm_monoid E]
-
-section ordered_add_comm_monoid
-variables [ordered_add_comm_monoid β]
-
-section has_scalar
-variables [has_scalar 𝕜 E] [has_scalar 𝕜 β] {s : set E}
-
-lemma convex_on_iff_div {f : E → β} :
-  convex_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
-  → f ((a/(a+b)) • x + (b/(a+b)) • y) ≤ (a/(a+b)) • f x + (b/(a+b)) • f y :=
-and_congr iff.rfl
-⟨begin
-  intros h x y hx hy a b ha hb hab,
-  apply h hx hy (div_nonneg ha $ le_of_lt hab) (div_nonneg hb $ le_of_lt hab),
-  rw [←add_div],
-  exact div_self (ne_of_gt hab)
-end,
-begin
-  intros h x y hx hy a b ha hb hab,
-  simpa [hab, zero_lt_one] using h hx hy ha hb,
-end⟩
-
-lemma concave_on_iff_div {f : E → β} :
-  concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
-  → (a/(a+b)) • f x + (b/(a+b)) • f y ≤ f ((a/(a+b)) • x + (b/(a+b)) • y) :=
-@convex_on_iff_div _ _ (order_dual β) _ _ _ _ _ _ _
-
-/-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
-the slope of the secant line of `f` on `[x, y]` is less than or equal to the slope
-of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of proving convexity
-of a function is used in the proof of convexity of a function with a monotone derivative. -/
-lemma convex_on_real_of_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜}
-  (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
-    (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :
-  convex_on 𝕜 s f :=
-linear_order.convex_on_of_lt hs
-begin
-  assume x z hx hz hxz a b ha hb hab,
-  let y := a * x + b * z,
-  have hxy : x < y,
-  { rw [← one_mul x, ← hab, add_mul],
-    exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },
-  have hyz : y < z,
-  { rw [← one_mul z, ← hab, add_mul],
-    exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ },
-  have : (f y - f x) * (z - y) ≤ (f z - f y) * (y - x),
-    from (div_le_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz),
-  have A : z - y + (y - x) = z - x, by abel,
-  have B : 0 < z - x, from sub_pos.2 (lt_trans hxy hyz),
-  rw [sub_mul, sub_mul, sub_le_iff_le_add', ← add_sub_assoc, le_sub_iff_add_le, ← mul_add, A,
-    ← le_div_iff B, add_div, mul_div_assoc, mul_div_assoc,
-    mul_comm (f x), mul_comm (f z)] at this,
-  rw [eq_comm, ← sub_eq_iff_eq_add] at hab; subst a,
-  convert this; symmetry; simp only [div_eq_iff (ne_of_gt B), y]; ring
-end
-
-/-- For a function `f` defined on a subset `D` of `𝕜`, if `f` is convex on `D`, then for any three
-points `x < y < z`, the slope of the secant line of `f` on `[x, y]` is less than or equal to the
-slope of the secant line of `f` on `[x, z]`. -/
-lemma convex_on.slope_mono_adjacent {s : set 𝕜} {f : 𝕜 → 𝕜} (hf : convex_on 𝕜 s f)
-  {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
-  (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) :=
-begin
-  have h₁ : 0 < y - x := by linarith,
-  have h₂ : 0 < z - y := by linarith,
-  have h₃ : 0 < z - x := by linarith,
-  suffices : f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y),
-    by { ring_nf at this ⊢, linarith },
-  set a := (z - y) / (z - x),
-  set b := (y - x) / (z - x),
-  have heqz : a • x + b • z = y, by { field_simp, rw div_eq_iff; [ring, linarith], },
-  have key, from
-    hf.2 hx hz
-      (show 0 ≤ a, by apply div_nonneg; linarith)
-      (show 0 ≤ b, by apply div_nonneg; linarith)
-      (show a + b = 1, by { field_simp, rw div_eq_iff; [ring, linarith], }),
-  rw heqz at key,
-  replace key := mul_le_mul_of_nonneg_left key (le_of_lt h₃),
-  field_simp [ne_of_gt h₁, ne_of_gt h₂, ne_of_gt h₃, mul_comm (z - x) _] at key ⊢,
-  rw div_le_div_right,
-  { linarith, },
-  { nlinarith, },
-end
-
-/-- For a function `f` defined on a convex subset `D` of `𝕜`, `f` is convex on `D` iff, for any
-three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is less than or equal to
-the slope,of the secant line of `f` on `[x, z]`. -/
-lemma convex_on_real_iff_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜} :
-  convex_on 𝕜 s f ↔
-  (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
-    (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :=
-⟨convex_on.slope_mono_adjacent, convex_on_real_of_slope_mono_adjacent hs⟩
+section module
+variables [module 𝕜 E] [module 𝕜 F] [has_scalar 𝕜 β]
 
-/-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
-the slope of the secant line of `f` on `[x, y]` is greater than or equal to the slope
-of the secant line of `f` on `[x, z]`, then `f` is concave on `D`. -/
-lemma concave_on_real_of_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜}
-  (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
-    (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : concave_on 𝕜 s f :=
-begin
-  rw [←neg_convex_on_iff],
-  apply convex_on_real_of_slope_mono_adjacent hs,
-  intros x y z xs zs xy yz,
-  rw [←neg_le_neg_iff, ←neg_div, ←neg_div, neg_sub, neg_sub],
-  simp only [hf xs zs xy yz, neg_sub_neg, pi.neg_apply],
-end
+/-- If `g` is convex on `s`, so is `(f ∘ g)` on `f ⁻¹' s` for a linear `f`. -/
+lemma convex_on.comp_linear_map {f : F → β} {s : set F} (hf : convex_on 𝕜 s f) (g : E →ₗ[𝕜] F) :
+  convex_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
+⟨hf.1.linear_preimage _, λ x y hx hy a b ha hb hab,
+  calc
+    f (g (a • x + b • y)) = f (a • (g x) + b • (g y)) : by rw [g.map_add, g.map_smul, g.map_smul]
+                      ... ≤ a • f (g x) + b • f (g y) : hf.2 hx hy ha hb hab⟩
 
-/-- For a function `f` defined on a subset `D` of `𝕜`, if `f` is concave on `D`, then for any three
-points `x < y < z`, the slope of the secant line of `f` on `[x, y]` is greater than or equal to the
-slope of the secant line of `f` on `[x, z]`. -/
-lemma concave_on.slope_mono_adjacent {s : set 𝕜} {f : 𝕜 → 𝕜} (hf : concave_on 𝕜 s f)
-  {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
-  (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) :=
-begin
-  rw [←neg_le_neg_iff, ←neg_div, ←neg_div, neg_sub, neg_sub],
-  rw [←neg_sub_neg (f y), ←neg_sub_neg (f z)],
-  simp_rw [←pi.neg_apply],
-  rw [←neg_convex_on_iff] at hf,
-  apply convex_on.slope_mono_adjacent hf; assumption,
-end
+/-- If `g` is concave on `s`, so is `(g ∘ f)` on `f ⁻¹' s` for a linear `f`. -/
+lemma concave_on.comp_linear_map {f : F → β} {s : set F} (hf : concave_on 𝕜 s f) (g : E →ₗ[𝕜] F) :
+  concave_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
+@convex_on.comp_linear_map 𝕜 E F (order_dual β) _ _ _ _ _ _ _ f s hf g
 
-/-- For a function `f` defined on a convex subset `D` of `𝕜`, `f` is concave on `D` iff for any
-three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is greater than or equal to
-the slope of the secant line of `f` on `[x, z]`. -/
-lemma concave_on_real_iff_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜} :
-  concave_on 𝕜 s f ↔
-  (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
-    (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) :=
-⟨concave_on.slope_mono_adjacent, concave_on_real_of_slope_mono_adjacent hs⟩
+end module
+end ordered_add_comm_monoid
 
-lemma convex_on.subset {f : E → β} {t : set E} (h_convex_on 𝕜 : convex_on 𝕜 t f)
-  (h_subset : s ⊆ t) (h_convex : convex 𝕜 s) : convex_on 𝕜 s f :=
-begin
-  apply and.intro h_convex,
-  intros x y hx hy,
-  exact h_convex_on.2 (h_subset hx) (h_subset hy),
-end
+section ordered_cancel_add_comm_monoid
+variables [ordered_cancel_add_comm_monoid β]
 
-lemma concave_on.subset {f : E → β} {t : set E} (h_concave_on 𝕜 : concave_on 𝕜 t f)
-  (h_subset : s ⊆ t) (h_convex : convex 𝕜 s) : concave_on 𝕜 s f :=
-@convex_on.subset _ (order_dual β) _ _ _ _ _ f t h_concave_on 𝕜 h_subset h_convex
+section module
+variables [module 𝕜 E] [module 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f g : E → β}
 
-lemma convex_on.add {f g : E → β} (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
-  convex_on 𝕜 s (λx, f x + g x) :=
+lemma convex_on.convex_lt (hf : convex_on 𝕜 s f) (r : β) : convex 𝕜 {x ∈ s | f x < r} :=
 begin
-  apply and.intro hf.1,
-  intros x y hx hy a b ha hb hab,
-  calc
-    f (a • x + b • y) + g (a • x + b • y) ≤ (a • f x + b • f y) + (a • g x + b • g y)
-      : add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)
-    ... = a • f x + a • g x + b • f y + b • g y : by abel
-    ... = a • (f x + g x) + b • (f y + g y) : by simp [smul_add, add_assoc]
+  refine λ x y hx hy a b ha hb hab, ⟨hf.1 hx.1 hy.1 ha hb hab, _⟩,
+  obtain rfl | ha' := ha.eq_or_lt,
+  { rw zero_add at hab,
+    rw [hab, zero_smul, one_smul, zero_add],
+    exact hy.2 },
+  { calc
+      f (a • x + b • y)
+          ≤ a • f x + b • f y : hf.2 hx.1 hy.1 ha hb hab
+      ... < a • r + b • r     : add_lt_add_of_lt_of_le (smul_lt_smul_of_pos hx.2 ha')
+                                  (smul_le_smul_of_nonneg hy.2.le hb)
+      ... = r                 : convex.combo_self hab _ }
 end
 
-lemma concave_on.add {f g : E → β} (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
-  concave_on 𝕜 s (λx, f x + g x) :=
-@convex_on.add _ (order_dual β) _ _ _ _ _ f g hf hg
-
-lemma convex_on.smul [ordered_smul 𝕜 β] {f : E → β} {c : 𝕜} (hc : 0 ≤ c)
-  (hf : convex_on 𝕜 s f) : convex_on 𝕜 s (λx, c • f x) :=
-begin
-  apply and.intro hf.1,
-  intros x y hx hy a b ha hb hab,
-  calc
-    c • f (a • x + b • y) ≤ c • (a • f x + b • f y)
-      : smul_le_smul_of_nonneg (hf.2 hx hy ha hb hab) hc
-    ... = a • (c • f x) + b • (c • f y) : by simp only [smul_add, smul_comm c]
-end
+lemma concave_on.convex_lt (hf : concave_on 𝕜 s f) (r : β) : convex 𝕜 {x ∈ s | r < f x} :=
+@convex_on.convex_lt 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf r
 
-lemma concave_on.smul [ordered_smul 𝕜 β] {f : E → β} {c : 𝕜} (hc : 0 ≤ c)
-  (hf : concave_on 𝕜 s f) : concave_on 𝕜 s (λx, c • f x) :=
-@convex_on.smul _ (order_dual β) _ _ _ _ _ _ f c hc hf
+end module
 
-section linear_order
-section monoid
+end ordered_cancel_add_comm_monoid
 
-variables {γ : Type*} [linear_ordered_add_comm_monoid γ] [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f g : E → γ}
+section linear_ordered_add_comm_monoid
+variables [linear_ordered_add_comm_monoid β] [has_scalar 𝕜 E] [module 𝕜 β] [ordered_smul 𝕜 β]
+  {s : set E} {f g : E → β}
 
 /-- The pointwise maximum of convex functions is convex. -/
 lemma convex_on.sup (hf : convex_on 𝕜 s f) (hg : convex_on 𝕜 s g) :
@@ -336,11 +274,11 @@ end
 /-- The pointwise minimum of concave functions is concave. -/
 lemma concave_on.inf (hf : concave_on 𝕜 s f) (hg : concave_on 𝕜 s g) :
   concave_on 𝕜 s (f ⊓ g) :=
-@convex_on.sup _ _ _ _ (order_dual γ) _ _ _ _ _ hf hg
+@convex_on.sup 𝕜 E (order_dual β) _ _ _ _ _ _ _ f g hf hg
 
 /-- A convex function on a segment is upper-bounded by the max of its endpoints. -/
-lemma convex_on.le_on_segment' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
-  (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
+lemma convex_on.le_on_segment' (hf : convex_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
+  {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
   f (a • x + b • y) ≤ max (f x) (f y) :=
 calc
   f (a • x + b • y) ≤ a • f x + b • f y : hf.2 hx hy ha hb hab
@@ -350,48 +288,46 @@ calc
   ... = max (f x) (f y) : by rw [←add_smul, hab, one_smul]
 
 /-- A concave function on a segment is lower-bounded by the min of its endpoints. -/
-lemma concave_on.le_on_segment' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
-  (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
+lemma concave_on.le_on_segment' (hf : concave_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
+ {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
   min (f x) (f y) ≤ f (a • x + b • y) :=
-@convex_on.le_on_segment' _ _ _ _ (order_dual γ) _ _ _ f hf x y a b hx hy ha hb hab
+@convex_on.le_on_segment' 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf x y hx hy a b ha hb hab
 
 /-- A convex function on a segment is upper-bounded by the max of its endpoints. -/
-lemma convex_on.le_on_segment (hf : convex_on 𝕜 s f) {x y z : E}
-  (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ [x -[𝕜] y]) :
+lemma convex_on.le_on_segment (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s) (hy : y ∈ s)
+  (hz : z ∈ [x -[𝕜] y]) :
   f z ≤ max (f x) (f y) :=
 let ⟨a, b, ha, hb, hab, hz⟩ := hz in hz ▸ hf.le_on_segment' hx hy ha hb hab
 
 /-- A concave function on a segment is lower-bounded by the min of its endpoints. -/
-lemma concave_on.le_on_segment {f : E → γ} (hf : concave_on 𝕜 s f) {x y z : E}
-  (hx : x ∈ s) (hy : y ∈ s) (hz : z ∈ [x -[𝕜] y]) :
-    min (f x) (f y) ≤ f z :=
-@convex_on.le_on_segment _ _ _ _ (order_dual γ) _ _ _ f hf x y z hx hy hz
+lemma concave_on.le_on_segment (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s) (hy : y ∈ s)
+  (hz : z ∈ [x -[𝕜] y]) :
+  min (f x) (f y) ≤ f z :=
+@convex_on.le_on_segment 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf x y z hx hy hz
+
+end linear_ordered_add_comm_monoid
 
-end monoid
+section linear_ordered_cancel_add_comm_monoid
+variables [linear_ordered_cancel_add_comm_monoid β]
 
-variables {γ : Type*} [linear_ordered_cancel_add_comm_monoid γ] [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ}
+section ordered_smul
+variables [has_scalar 𝕜 E] [module 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f g : E → β}
 
--- could be shown without contradiction but yeah
-lemma convex_on.le_left_of_right_le' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
-  (hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
-  (hxy : f y ≤ f (a • x + b • y)) :
+lemma convex_on.le_left_of_right_le' (hf : convex_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
+ {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) (hxy : f y ≤ f (a • x + b • y)) :
   f (a • x + b • y) ≤ f x :=
-begin
-  apply le_of_not_lt (λ h, lt_irrefl (f (a • x + b • y)) _),
+le_of_not_lt $ λ h, lt_irrefl (f (a • x + b • y)) $
   calc
     f (a • x + b • y)
         ≤ a • f x + b • f y : hf.2 hx hy ha.le hb hab
     ... < a • f (a • x + b • y) + b • f (a • x + b • y)
         : add_lt_add_of_lt_of_le (smul_lt_smul_of_pos h ha) (smul_le_smul_of_nonneg hxy hb)
-    ... = f (a • x + b • y) : by rw [←add_smul, hab, one_smul],
-end
+    ... = f (a • x + b • y) : by rw [←add_smul, hab, one_smul]
 
-lemma concave_on.left_le_of_le_right' (hf : concave_on 𝕜 s f) {x y : E} {a b : 𝕜}
-  (hx : x ∈ s) (hy : y ∈ s) (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
-  (hxy : f (a • x + b • y) ≤ f y) :
+lemma concave_on.left_le_of_le_right' (hf : concave_on 𝕜 s f) {x y : E} (hx : x ∈ s) (hy : y ∈ s)
+  {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) (hxy : f (a • x + b • y) ≤ f y) :
   f x ≤ f (a • x + b • y) :=
-@convex_on.le_left_of_right_le' _ _ _ _ (order_dual γ) _ _ _ f hf x y a b hx hy ha hb hab hxy
+@convex_on.le_left_of_right_le' 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf x y hx hy a b ha hb hab hxy
 
 lemma convex_on.le_right_of_left_le' (hf : convex_on 𝕜 s f) {x y : E} {a b : 𝕜}
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
@@ -406,7 +342,7 @@ lemma concave_on.le_right_of_left_le' (hf : concave_on 𝕜 s f) {x y : E} {a b
   (hx : x ∈ s) (hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
   (hxy : f (a • x + b • y) ≤ f x) :
   f y ≤ f (a • x + b • y) :=
-@convex_on.le_right_of_left_le' _ _ _ _ (order_dual γ) _ _ _ f hf x y a b hx hy ha hb hab hxy
+@convex_on.le_right_of_left_le' 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf x y a b hx hy ha hb hab hxy
 
 lemma convex_on.le_left_of_right_le (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
   (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f y ≤ f z) :
@@ -419,7 +355,7 @@ end
 lemma concave_on.left_le_of_le_right (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
   (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hyz : f z ≤ f y) :
   f x ≤ f z :=
-@convex_on.le_left_of_right_le _ _ _ _ (order_dual γ) _ _ _ f hf x y z hx hy hz hyz
+@convex_on.le_left_of_right_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf x y z hx hy hz hyz
 
 lemma convex_on.le_right_of_left_le (hf : convex_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
   (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f x ≤ f z) :
@@ -432,168 +368,285 @@ end
 lemma concave_on.le_right_of_left_le (hf : concave_on 𝕜 s f) {x y z : E} (hx : x ∈ s)
   (hy : y ∈ s) (hz : z ∈ open_segment 𝕜 x y) (hxz : f z ≤ f x) :
   f y ≤ f z :=
-@convex_on.le_right_of_left_le _ _ _ _ (order_dual γ) _ _ _ f hf x y z hx hy hz hxz
+@convex_on.le_right_of_left_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf x y z hx hy hz hxz
 
-end linear_order
+end ordered_smul
+end linear_ordered_cancel_add_comm_monoid
 
-lemma convex_on.convex_le [ordered_smul 𝕜 β] {f : E → β} (hf : convex_on 𝕜 s f) (r : β) :
-  convex 𝕜 {x ∈ s | f x ≤ r} :=
-λ x y hx hy a b ha hb hab,
+section ordered_add_comm_group
+variables [ordered_add_comm_group β] [has_scalar 𝕜 E] [module 𝕜 β] {s : set E} {f : E → β}
+
+/-- A function `-f` is convex iff `f` is concave. -/
+@[simp] lemma neg_convex_on_iff :
+  convex_on 𝕜 s (-f) ↔ concave_on 𝕜 s f :=
 begin
-  refine ⟨hf.1 hx.1 hy.1 ha hb hab, _⟩,
-  calc
-    f (a • x + b • y) ≤ a • (f x) + b • (f y) : hf.2 hx.1 hy.1 ha hb hab
-                  ... ≤ a • r + b • r         : add_le_add (smul_le_smul_of_nonneg hx.2 ha)
-                                                  (smul_le_smul_of_nonneg hy.2 hb)
-                  ... ≤ r                     : by simp [←add_smul, hab]
+  split,
+  { rintro ⟨hconv, h⟩,
+    refine ⟨hconv, λ x y hx hy a b ha hb hab, _⟩,
+    simp [neg_apply, neg_le, add_comm] at h,
+    exact h hx hy ha hb hab },
+  { rintro ⟨hconv, h⟩,
+    refine ⟨hconv, λ x y hx hy a b ha hb hab, _⟩,
+    rw ←neg_le_neg_iff,
+    simp_rw [neg_add, pi.neg_apply, smul_neg, neg_neg],
+    exact h hx hy ha hb hab }
 end
 
-lemma concave_on.concave_le [ordered_smul 𝕜 β] {f : E → β} (hf : concave_on 𝕜 s f) (r : β) :
-  convex 𝕜 {x ∈ s | r ≤ f x} :=
-@convex_on.convex_le _ (order_dual β) _ _ _ _ _ _ f hf r
+/-- A function `-f` is concave iff `f` is convex. -/
+@[simp] lemma neg_concave_on_iff : concave_on 𝕜 s (-f) ↔ convex_on 𝕜 s f:=
+by rw [← neg_convex_on_iff, neg_neg f]
 
-lemma convex_on.convex_lt {γ : Type*} [ordered_cancel_add_comm_monoid γ]
-  [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ} (hf : convex_on 𝕜 s f) (r : γ) : convex 𝕜 {x ∈ s | f x < r} :=
-begin
-  intros a b as bs xa xb hxa hxb hxaxb,
-  refine ⟨hf.1 as.1 bs.1 hxa hxb hxaxb, _⟩,
-  by_cases H : xa = 0,
-  { have H' : xb = 1 := by rwa [H, zero_add] at hxaxb,
-    rw [H, H', zero_smul, one_smul, zero_add],
-    exact bs.2 },
-  { calc
-      f (xa • a + xb • b) ≤ xa • (f a) + xb • (f b) : hf.2 as.1 bs.1 hxa hxb hxaxb
-                      ... < xa • r + xb • (f b)     : (add_lt_add_iff_right (xb • (f b))).mpr
-                                                        (smul_lt_smul_of_pos as.2
-                                                          (lt_of_le_of_ne hxa (ne.symm H)))
-                      ... ≤ xa • r + xb • r         : (add_le_add_iff_left (xa • r)).mpr
-                                                        (smul_le_smul_of_nonneg bs.2.le hxb)
-                      ... = r                       : by simp only [←add_smul, hxaxb, one_smul] }
-end
+alias neg_convex_on_iff ↔ _ concave_on.neg
+alias neg_concave_on_iff ↔ _ convex_on.neg
 
-lemma concave_on.convex_lt {γ : Type*} [ordered_cancel_add_comm_monoid γ]
-  [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ} (hf : concave_on 𝕜 s f) (r : γ) : convex 𝕜 {x ∈ s | r < f x} :=
-@convex_on.convex_lt _ _ _ _ (order_dual γ) _ _ _ f hf r
+end ordered_add_comm_group
+end ordered_semiring
 
-lemma convex_on.convex_epigraph {γ : Type*} [ordered_add_comm_group γ]
-  [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ} (hf : convex_on 𝕜 s f) :
-  convex 𝕜 {p : E × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
-begin
-  rintros ⟨x, r⟩ ⟨y, t⟩ ⟨hx, hr⟩ ⟨hy, ht⟩ a b ha hb hab,
-  refine ⟨hf.1 hx hy ha hb hab, _⟩,
-  calc f (a • x + b • y) ≤ a • f x + b • f y : hf.2 hx hy ha hb hab
-  ... ≤ a • r + b • t : add_le_add (smul_le_smul_of_nonneg hr ha)
-                            (smul_le_smul_of_nonneg ht hb)
-end
+section ordered_comm_semiring
+variables [ordered_comm_semiring 𝕜] [add_comm_monoid E]
 
-lemma concave_on.convex_hypograph {γ : Type*} [ordered_add_comm_group γ]
-  [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ} (hf : concave_on 𝕜 s f) :
-  convex 𝕜 {p : E × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
-@convex_on.convex_epigraph _ _ _ _ (order_dual γ) _ _ _ f hf
+section ordered_add_comm_monoid
+variables [ordered_add_comm_monoid β]
 
-lemma convex_on_iff_convex_epigraph {γ : Type*} [ordered_add_comm_group γ]
-  [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ} :
-  convex_on 𝕜 s f ↔ convex 𝕜 {p : E × γ | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
-begin
-  refine ⟨convex_on.convex_epigraph, λ h, ⟨_, _⟩⟩,
-  { assume x y hx hy a b ha hb hab,
-    exact (@h (x, f x) (y, f y) ⟨hx, le_refl _⟩ ⟨hy, le_refl _⟩ a b ha hb hab).1 },
-  { assume x y hx hy a b ha hb hab,
-    exact (@h (x, f x) (y, f y) ⟨hx, le_refl _⟩ ⟨hy, le_refl _⟩ a b ha hb hab).2 }
-end
+section module
+variables [has_scalar 𝕜 E] [module 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f : E → β}
 
-lemma concave_on_iff_convex_hypograph {γ : Type*} [ordered_add_comm_group γ]
-  [module 𝕜 γ] [ordered_smul 𝕜 γ]
-  {f : E → γ} :
-  concave_on 𝕜 s f ↔ convex 𝕜 {p : E × γ | p.1 ∈ s ∧ p.2 ≤ f p.1} :=
-@convex_on_iff_convex_epigraph _ _ _ _ (order_dual γ) _ _ _ f
+lemma convex_on.smul {c : 𝕜} (hc : 0 ≤ c)
+  (hf : convex_on 𝕜 s f) : convex_on 𝕜 s (λ x, c • f x) :=
+⟨hf.1, λ x y hx hy a b ha hb hab,
+  calc
+    c • f (a • x + b • y) ≤ c • (a • f x + b • f y)
+      : smul_le_smul_of_nonneg (hf.2 hx hy ha hb hab) hc
+    ... = a • (c • f x) + b • (c • f y)
+      : by rw [smul_add, smul_comm c, smul_comm c]; apply_instance⟩
 
-/- A linear map is convex. -/
-lemma linear_map.convex_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : convex_on 𝕜 s f :=
-⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩
+lemma concave_on.smul {c : 𝕜} (hc : 0 ≤ c)
+  (hf : concave_on 𝕜 s f) : concave_on 𝕜 s (λ x, c • f x) :=
+@convex_on.smul _ _ (order_dual β) _ _ _ _ _ _ _ f c hc hf
 
-/- A linear map is concave. -/
-lemma linear_map.concave_on (f : E →ₗ[𝕜] β) {s : set E} (hs : convex 𝕜 s) : concave_on 𝕜 s f :=
-⟨hs, λ _ _ _ _ _ _ _ _ _, by rw [f.map_add, f.map_smul, f.map_smul]⟩
+end module
+end ordered_add_comm_monoid
+end ordered_comm_semiring
+
+section ordered_ring
+variables [linear_ordered_field 𝕜] [add_comm_group E] [add_comm_group F]
+
+section ordered_add_comm_monoid
+variables [ordered_add_comm_monoid β]
+
+section module
+variables [module 𝕜 E] [module 𝕜 F] [has_scalar 𝕜 β]
 
 /-- If a function is convex on `s`, it remains convex when precomposed by an affine map. -/
-lemma convex_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F}
-  (hf : convex_on 𝕜 s f) : convex_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
-begin
-  refine ⟨hf.1.affine_preimage  _,_⟩,
-  intros x y xs ys a b ha hb hab,
+lemma convex_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F} (hf : convex_on 𝕜 s f) :
+  convex_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
+⟨hf.1.affine_preimage _, λ x y hx hy a b ha hb hab,
   calc
     (f ∘ g) (a • x + b • y) = f (g (a • x + b • y))         : rfl
                        ...  = f (a • (g x) + b • (g y))     : by rw [convex.combo_affine_apply hab]
-                       ...  ≤ a • f (g x) + b • f (g y)     : hf.2 xs ys ha hb hab
-                       ...  = a • (f ∘ g) x + b • (f ∘ g) y : rfl
-end
+                       ...  ≤ a • f (g x) + b • f (g y)     : hf.2 hx hy ha hb hab⟩
 
 /-- If a function is concave on `s`, it remains concave when precomposed by an affine map. -/
-lemma concave_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F}
-  (hf : concave_on 𝕜 s f) : concave_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
-@convex_on.comp_affine_map _ _ (order_dual β) _ _ _ _ _ _ f g s hf
+lemma concave_on.comp_affine_map {f : F → β} (g : E →ᵃ[𝕜] F) {s : set F} (hf : concave_on 𝕜 s f) :
+  concave_on 𝕜 (g ⁻¹' s) (f ∘ g) :=
+@convex_on.comp_affine_map _ _ _ (order_dual β) _ _ _ _ _ _ _ f g s hf
+
+end module
+end ordered_add_comm_monoid
+end ordered_ring
+
+section linear_ordered_field
+variables [linear_ordered_field 𝕜] [add_comm_monoid E]
+
+section ordered_add_comm_monoid
+variables [ordered_add_comm_monoid β]
+
+section has_scalar
+variables [has_scalar 𝕜 E] [has_scalar 𝕜 β] {s : set E}
+
+lemma convex_on_iff_div {f : E → β} :
+  convex_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
+  → f ((a/(a+b)) • x + (b/(a+b)) • y) ≤ (a/(a+b)) • f x + (b/(a+b)) • f y :=
+and_congr iff.rfl
+⟨begin
+  intros h x y hx hy a b ha hb hab,
+  apply h hx hy (div_nonneg ha hab.le) (div_nonneg hb hab.le),
+  rw [←add_div, div_self hab.ne'],
+end,
+begin
+  intros h x y hx hy a b ha hb hab,
+  simpa [hab, zero_lt_one] using h hx hy ha hb,
+end⟩
+
+lemma concave_on_iff_div {f : E → β} :
+  concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
+  → (a/(a+b)) • f x + (b/(a+b)) • f y ≤ f ((a/(a+b)) • x + (b/(a+b)) • y) :=
+@convex_on_iff_div _ _ (order_dual β) _ _ _ _ _ _ _
+
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
+the slope of the secant line of `f` on `[x, y]` is less than or equal to the slope
+of the secant line of `f` on `[x, z]`, then `f` is convex on `D`. This way of proving convexity
+of a function is used in the proof of convexity of a function with a monotone derivative. -/
+lemma convex_on_of_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜}
+  (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
+    (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :
+  convex_on 𝕜 s f :=
+linear_order.convex_on_of_lt hs
+begin
+  assume x z hx hz hxz a b ha hb hab,
+  let y := a * x + b * z,
+  have hxy : x < y,
+  { rw [← one_mul x, ← hab, add_mul],
+    exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _ },
+  have hyz : y < z,
+  { rw [← one_mul z, ← hab, add_mul],
+    exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _ },
+  have : (f y - f x) * (z - y) ≤ (f z - f y) * (y - x),
+    from (div_le_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz),
+  have hxz : 0 < z - x, from sub_pos.2 (hxy.trans hyz),
+  have ha : (z - y) / (z - x) = a,
+  { rw [eq_comm, ← sub_eq_iff_eq_add'] at hab,
+    simp_rw [div_eq_iff hxz.ne', y, ←hab], ring },
+  have hb : (y - x) / (z - x) = b,
+  { rw [eq_comm, ← sub_eq_iff_eq_add] at hab,
+    simp_rw [div_eq_iff hxz.ne', y, ←hab], ring },
+  rwa [sub_mul, sub_mul, sub_le_iff_le_add', ← add_sub_assoc, le_sub_iff_add_le, ← mul_add,
+    sub_add_sub_cancel, ← le_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x),
+    mul_comm (f z), ha, hb] at this,
+end
+
+/-- For a function `f` defined on a subset `D` of `𝕜`, if `f` is convex on `D`, then for any three
+points `x < y < z`, the slope of the secant line of `f` on `[x, y]` is less than or equal to the
+slope of the secant line of `f` on `[x, z]`. -/
+lemma convex_on.slope_mono_adjacent {s : set 𝕜} {f : 𝕜 → 𝕜} (hf : convex_on 𝕜 s f)
+  {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
+  (f y - f x) / (y - x) ≤ (f z - f y) / (z - y) :=
+begin
+  have h₁ : 0 < y - x := by linarith,
+  have h₂ : 0 < z - y := by linarith,
+  have h₃ : 0 < z - x := by linarith,
+  suffices : f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y),
+  { ring_nf at this ⊢, linarith },
+  set a := (z - y) / (z - x),
+  set b := (y - x) / (z - x),
+  have heqz : a • x + b • z = y, by { field_simp, rw div_eq_iff; [ring, linarith] },
+  have key, from
+    hf.2 hx hz
+      (show 0 ≤ a, by apply div_nonneg; linarith)
+      (show 0 ≤ b, by apply div_nonneg; linarith)
+      (show a + b = 1, by { field_simp, rw div_eq_iff; [ring, linarith] }),
+  rw heqz at key,
+  replace key := mul_le_mul_of_nonneg_left key h₃.le,
+  field_simp [h₁.ne', h₂.ne', h₃.ne', mul_comm (z - x) _] at key ⊢,
+  rw div_le_div_right,
+  { linarith },
+  { nlinarith }
+end
+
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, `f` is convex on `D` iff, for any
+three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is less than or equal to
+the slope,of the secant line of `f` on `[x, z]`. -/
+lemma convex_on_iff_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜} :
+  convex_on 𝕜 s f ↔
+  (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
+    (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)) :=
+⟨convex_on.slope_mono_adjacent, convex_on_of_slope_mono_adjacent hs⟩
+
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
+the slope of the secant line of `f` on `[x, y]` is greater than or equal to the slope
+of the secant line of `f` on `[x, z]`, then `f` is concave on `D`. -/
+lemma concave_on_of_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜}
+  (hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
+    (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) : concave_on 𝕜 s f :=
+begin
+  rw ←neg_convex_on_iff,
+  refine convex_on_of_slope_mono_adjacent hs (λ x y z hx hz hxy hyz, _),
+  rw ←neg_le_neg_iff,
+  simp_rw [←neg_div, neg_sub, pi.neg_apply, neg_sub_neg],
+  exact hf hx hz hxy hyz,
+end
+
+/-- For a function `f` defined on a subset `D` of `𝕜`, if `f` is concave on `D`, then for any three
+points `x < y < z`, the slope of the secant line of `f` on `[x, y]` is greater than or equal to the
+slope of the secant line of `f` on `[x, z]`. -/
+lemma concave_on.slope_mono_adjacent {s : set 𝕜} {f : 𝕜 → 𝕜} (hf : concave_on 𝕜 s f)
+  {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s) (hxy : x < y) (hyz : y < z) :
+  (f z - f y) / (z - y) ≤ (f y - f x) / (y - x) :=
+begin
+  rw [←neg_le_neg_iff, ←neg_sub_neg (f x), ←neg_sub_neg (f y)],
+  simp_rw [←pi.neg_apply, ←neg_div, neg_sub],
+  exact convex_on.slope_mono_adjacent hf.neg hx hz hxy hyz,
+end
+
+/-- For a function `f` defined on a convex subset `D` of `𝕜`, `f` is concave on `D` iff for any
+three points `x < y < z` the slope of the secant line of `f` on `[x, y]` is greater than or equal to
+the slope of the secant line of `f` on `[x, z]`. -/
+lemma concave_on_iff_slope_mono_adjacent {s : set 𝕜} (hs : convex 𝕜 s) {f : 𝕜 → 𝕜} :
+  concave_on 𝕜 s f ↔
+  (∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z →
+    (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)) :=
+⟨concave_on.slope_mono_adjacent, concave_on_of_slope_mono_adjacent hs⟩
+
+end has_scalar
+end ordered_add_comm_monoid
+end linear_ordered_field
 
-/-- If `g` is convex on `s`, so is `(g ∘ f)` on `f ⁻¹' s` for a linear `f`. -/
-lemma convex_on.comp_linear_map {g : F → β} {s : set F} (hg : convex_on 𝕜 s g) (f : E →ₗ[𝕜] F) :
-  convex_on 𝕜 (f ⁻¹' s) (g ∘ f) :=
-hg.comp_affine_map f.to_affine_map
 
-/-- If `g` is concave on `s`, so is `(g ∘ f)` on `f ⁻¹' s` for a linear `f`. -/
-lemma concave_on.comp_linear_map {g : F → β} {s : set F} (hg : concave_on 𝕜 s g) (f : E →ₗ[𝕜] F) :
-  concave_on 𝕜 (f ⁻¹' s) (g ∘ f) :=
-hg.comp_affine_map f.to_affine_map
 
-/-- If a function is convex on `s`, it remains convex after a translation. -/
-lemma convex_on.translate_right {f : E → β} {s : set E} {a : E} (hf : convex_on 𝕜 s f) :
-  convex_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
-hf.comp_affine_map $ affine_map.const 𝕜 E a +ᵥ affine_map.id 𝕜 E
 
-/-- If a function is concave on `s`, it remains concave after a translation. -/
-lemma concave_on.translate_right {f : E → β} {s : set E} {a : E} (hf : concave_on 𝕜 s f) :
-  concave_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, a + z)) :=
-hf.comp_affine_map $ affine_map.const 𝕜 E a +ᵥ affine_map.id 𝕜 E
 
-/-- If a function is convex on `s`, it remains convex after a translation. -/
-lemma convex_on.translate_left {f : E → β} {s : set E} {a : E} (hf : convex_on 𝕜 s f) :
-  convex_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
-by simpa only [add_comm] using hf.translate_right
 
-/-- If a function is concave on `s`, it remains concave after a translation. -/
-lemma concave_on.translate_left {f : E → β} {s : set E} {a : E} (hf : concave_on 𝕜 s f) :
-  concave_on 𝕜 ((λ z, a + z) ⁻¹' s) (f ∘ (λ z, z + a)) :=
-by simpa only [add_comm] using hf.translate_right
 
 /-! ### Jensen's inequality -/
 
-variables {i j : ι} {c : 𝕜} {t : finset ι} {w : ι → 𝕜} {p : ι → E}
+section jensen
+variables [linear_ordered_field 𝕜] [add_comm_monoid E] [ordered_add_comm_monoid β] [module 𝕜 E]
+  [module 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f : E → β} {t : finset ι} {w : ι → 𝕜} {p : ι → E}
 
 /-- Convex **Jensen's inequality**, `finset.linear_combination` version. -/
-lemma convex_on.map_linear_combination_le {f : E → 𝕜} (hf : convex_on 𝕜 s f)
-  (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
-  (hmem : ∀ i ∈ t, p i ∈ s) : f (t.linear_combination p w) ≤ t.linear_combination (f ∘ p) w :=
+lemma convex_on.map_linear_combination_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
+  (h₁ : ∑ i in t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
+  f (t.linear_combination p w) ≤ t.linear_combination (f ∘ p) w :=
 begin
-  have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × 𝕜 | p.1 ∈ s ∧ f p.1 ≤ p.2},
+  have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2},
     from λ i hi, ⟨hmem i hi, le_rfl⟩,
   convert (hf.convex_epigraph.linear_combination_mem h₀ h₁ hmem').2;
     simp only [linear_combination, function.comp, prod.smul_fst, prod.fst_sum,
-      prod.smul_snd, prod.snd_sum]
+      prod.smul_snd, prod.snd_sum],
 end
 
+/-- Concave **Jensen's inequality**, `finset.linear_combination` version. -/
+lemma concave_on.le_map_linear_combination (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
+  (h₁ : ∑ i in t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
+  t.linear_combination (f ∘ p) w ≤ f (t.linear_combination p w) :=
+@convex_on.map_linear_combination_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ _ _ _ _ hf h₀ h₁ hmem
+
 /-- Convex **Jensen's inequality**, `finset.sum` version. -/
-lemma convex_on.map_sum_le {f : E → 𝕜} (hf : convex_on 𝕜 s f)
-  (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
-  (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i in t, w i • p i) ≤ ∑ i in t, w i * (f (p i)) :=
-by simpa only [linear_combination, h₁, inv_one, one_smul]
-  using hf.map_linear_combination_le h₀ h₁ hmem
+lemma convex_on.map_sum_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i ≠ 0)
+  (hmem : ∀ i ∈ t, p i ∈ s) :
+  f ((∑ i in t, w i)⁻¹ • ∑ i in t, w i • p i) ≤ (∑ i in t, w i)⁻¹ • ∑ i in t, w i • f (p i) :=
+begin
+  rw [←linear_combination, ←linear_combination_smul_right, ←linear_combination,
+    ←linear_combination_smul_right],
+  exact hf.map_linear_combination_le (linear_combination_normalize_weight_nonneg h₀)
+    (linear_combination_normalize_weight_sum h₁) hmem,
+end
+
+/-- Concave **Jensen's inequality**, `finset.sum` version. -/
+lemma concave_on.le_map_sum (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i ≠ 0)
+  (hmem : ∀ i ∈ t, p i ∈ s) :
+  (∑ i in t, w i)⁻¹ • ∑ i in t, w i • f (p i) ≤ f ((∑ i in t, w i)⁻¹ • ∑ i in t, w i • p i) :=
+@convex_on.map_sum_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ _ _ _ _ hf h₀ h₁ hmem
+
+end jensen
 
 /-! ### Maximum principle -/
 
+section maximum_principle
+variables [linear_ordered_field 𝕜] [add_comm_monoid E] [ordered_add_comm_monoid β] [module 𝕜 E]
+  [mul_action_with_zero 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f : E → β} {t : finset ι} {w : ι → 𝕜}
+  {p : ι → E}
+
 /-- If a function `f` is convex on `s` takes value `y` at the center of mass of some points
 `p i ∈ s`, then for some `i` we have `y ≤ f (p i)`. -/
 lemma convex_on.exists_ge_of_linear_combination {f : E → 𝕜} (h : convex_on 𝕜 s f)
@@ -625,3 +678,5 @@ begin
     (λ i hi, subset_convex_hull 𝕜 s (hp i hi)) with ⟨i, hit, Hi⟩,
   exact ⟨p i, hp i hit, Hi⟩
 end
+
+end maximum_principle
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index 793bbe5e9e42f..0bee342349f07 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -21,13 +21,14 @@ In this file we prove four theorems:
   belongs to `s`: `∫ x, f x ∂μ ∈ s`. See also `convex.linear_combination_mem` for a finite sum
   version of this lemma.
 
-* `convex_on.map_smul_integral_le`: Jensen's inequality: if a function `g : E → ℝ` is convex and
-  continuous on a convex closed set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is
-  a function sending `μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is
-  less than or equal to the average value of `g ∘ f` provided that both `f` and `g ∘ f` are
-  integrable. See also `convex.map_linear_combination_le` for a finite sum version of this lemma.
+* `convex_on.map_smul_integral_le`: Convex Jensen's inequality: If a function `g : E → ℝ` is convex
+  and continuous on a convex closed set `s`, `μ` is a finite non-zero measure on `α`, and
+  `f : α → E` is a function sending `μ`-a.e. points to `s`, then the value of `g` at the average
+  value of `f` is less than or equal to the average value of `g ∘ f` provided that both `f` and
+  `g ∘ f` are integrable. See also `convex_on.map_linear_combination_le` for a finite sum version of
+  this lemma.
 
-* `convex_on.map_integral_le`: Jensen's inequality: if a function `g : E → ℝ` is convex and
+* `convex_on.map_integral_le`: Convex Jensen's inequality: If a function `g : E → ℝ` is convex and
   continuous on a convex closed set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a
   function sending `μ`-a.e. points to `s`, then the value of `g` at the expected value of `f` is
   less than or equal to the expected value of `g ∘ f` provided that both `f` and `g ∘ f` are

From 410eb13a3d98b455e6b397e89d2db0fcdbdabfe7 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Thu, 23 Sep 2021 16:14:10 +0200
Subject: [PATCH 11/17] other files

---
 src/analysis/calculus/mean_value.lean       | 23 ++++++++++-----------
 src/analysis/convex/exposed.lean            | 10 ++++-----
 src/analysis/convex/extrema.lean            | 10 ++++-----
 src/analysis/convex/function.lean           | 10 ++++-----
 src/analysis/convex/integral.lean           |  4 ++--
 src/analysis/convex/specific_functions.lean | 16 +++++++-------
 src/analysis/convex/topology.lean           |  2 +-
 7 files changed, 37 insertions(+), 38 deletions(-)

diff --git a/src/analysis/calculus/mean_value.lean b/src/analysis/calculus/mean_value.lean
index dbf910b1fa619..bfae9684e62d9 100644
--- a/src/analysis/calculus/mean_value.lean
+++ b/src/analysis/calculus/mean_value.lean
@@ -973,7 +973,7 @@ and `f'` is monotone on the interior, then `f` is convex on `D`. -/
 theorem convex_on_of_deriv_mono {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
   (hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
   (hf'_mono : ∀ x y ∈ interior D, x ≤ y → deriv f x ≤ deriv f y) :
-  convex_on D f :=
+  convex_on ℝ D f :=
 convex_on_of_slope_mono_adjacent hD
 begin
   intros x y z hx hz hxy hyz,
@@ -999,25 +999,24 @@ and `f'` is antimonotone on the interior, then `f` is concave on `D`. -/
 theorem concave_on_of_deriv_antimono {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
   (hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
   (hf'_mono : ∀ x y ∈ interior D, x ≤ y → deriv f y ≤ deriv f x) :
-  concave_on D f :=
+  concave_on ℝ D f :=
 begin
   have : ∀ x y ∈ interior D, x ≤ y → deriv (-f) x ≤ deriv (-f) y,
   { intros x y hx hy hxy,
     convert neg_le_neg (hf'_mono x y hx hy hxy);
     convert deriv.neg },
-  exact (neg_convex_on_iff D f).mp (convex_on_of_deriv_mono hD
-    hf.neg hf'.neg this),
+  exact neg_convex_on_iff.mp (convex_on_of_deriv_mono hD hf.neg hf'.neg this),
 end
 
 /-- If a function `f` is differentiable and `f'` is monotone on `ℝ` then `f` is convex. -/
 theorem convex_on_univ_of_deriv_mono {f : ℝ → ℝ} (hf : differentiable ℝ f)
-  (hf'_mono : monotone (deriv f)) : convex_on univ f :=
+  (hf'_mono : monotone (deriv f)) : convex_on ℝ univ f :=
 convex_on_of_deriv_mono convex_univ hf.continuous.continuous_on hf.differentiable_on
   (λ x y _ _ h, hf'_mono h)
 
 /-- If a function `f` is differentiable and `f'` is antimonotone on `ℝ` then `f` is concave. -/
 theorem concave_on_univ_of_deriv_antimono {f : ℝ → ℝ} (hf : differentiable ℝ f)
-  (hf'_antimono : ∀⦃a b⦄, a ≤ b → (deriv f) b ≤ (deriv f) a) : concave_on univ f :=
+  (hf'_antimono : ∀⦃a b⦄, a ≤ b → (deriv f) b ≤ (deriv f) a) : concave_on ℝ univ f :=
 concave_on_of_deriv_antimono convex_univ hf.continuous.continuous_on hf.differentiable_on
   (λ x y _ _ h, hf'_antimono h)
 
@@ -1027,7 +1026,7 @@ theorem convex_on_of_deriv2_nonneg {D : set ℝ} (hD : convex ℝ D) {f : ℝ 
   (hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
   (hf'' : differentiable_on ℝ (deriv f) (interior D))
   (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ (deriv^[2] f x)) :
-  convex_on D f :=
+  convex_on ℝ D f :=
 convex_on_of_deriv_mono hD hf hf' $
 assume x y hx hy hxy,
 hD.interior.mono_of_deriv_nonneg hf''.continuous_on (by rwa [interior_interior])
@@ -1039,7 +1038,7 @@ theorem concave_on_of_deriv2_nonpos {D : set ℝ} (hD : convex ℝ D) {f : ℝ 
   (hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
   (hf'' : differentiable_on ℝ (deriv f) (interior D))
   (hf''_nonpos : ∀ x ∈ interior D, (deriv^[2] f x) ≤ 0) :
-  concave_on D f :=
+  concave_on ℝ D f :=
 concave_on_of_deriv_antimono hD hf hf' $
 assume x y hx hy hxy,
 hD.interior.antimono_of_deriv_nonpos hf''.continuous_on (by rwa [interior_interior])
@@ -1049,7 +1048,7 @@ hD.interior.antimono_of_deriv_nonpos hf''.continuous_on (by rwa [interior_interi
 `f''` is nonnegative on `D`, then `f` is convex on `D`. -/
 theorem convex_on_open_of_deriv2_nonneg {D : set ℝ} (hD : convex ℝ D) (hD₂ : is_open D) {f : ℝ → ℝ}
   (hf' : differentiable_on ℝ f D) (hf'' : differentiable_on ℝ (deriv f) D)
-  (hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f x)) : convex_on D f :=
+  (hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f x)) : convex_on ℝ D f :=
 convex_on_of_deriv2_nonneg hD hf'.continuous_on (by simpa [hD₂.interior_eq] using hf')
   (by simpa [hD₂.interior_eq] using hf'') (by simpa [hD₂.interior_eq] using hf''_nonneg)
 
@@ -1057,7 +1056,7 @@ convex_on_of_deriv2_nonneg hD hf'.continuous_on (by simpa [hD₂.interior_eq] us
 `f''` is nonpositive on `D`, then `f` is concave on `D`. -/
 theorem concave_on_open_of_deriv2_nonpos {D : set ℝ} (hD : convex ℝ D) (hD₂ : is_open D) {f : ℝ → ℝ}
   (hf' : differentiable_on ℝ f D) (hf'' : differentiable_on ℝ (deriv f) D)
-  (hf''_nonpos : ∀ x ∈ D, (deriv^[2] f x) ≤ 0) : concave_on D f :=
+  (hf''_nonpos : ∀ x ∈ D, (deriv^[2] f x) ≤ 0) : concave_on ℝ D f :=
 concave_on_of_deriv2_nonpos hD hf'.continuous_on (by simpa [hD₂.interior_eq] using hf')
   (by simpa [hD₂.interior_eq] using hf'') (by simpa [hD₂.interior_eq] using hf''_nonpos)
 
@@ -1065,7 +1064,7 @@ concave_on_of_deriv2_nonpos hD hf'.continuous_on (by simpa [hD₂.interior_eq] u
 then `f` is convex on `ℝ`. -/
 theorem convex_on_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : differentiable ℝ f)
   (hf'' : differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ (deriv^[2] f x)) :
-  convex_on univ f :=
+  convex_on ℝ univ f :=
 convex_on_open_of_deriv2_nonneg convex_univ is_open_univ hf'.differentiable_on
   hf''.differentiable_on (λ x _, hf''_nonneg x)
 
@@ -1073,7 +1072,7 @@ convex_on_open_of_deriv2_nonneg convex_univ is_open_univ hf'.differentiable_on
 then `f` is concave on `ℝ`. -/
 theorem concave_on_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : differentiable ℝ f)
   (hf'' : differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, (deriv^[2] f x) ≤ 0) :
-  concave_on univ f :=
+  concave_on ℝ univ f :=
 concave_on_open_of_deriv2_nonpos convex_univ is_open_univ hf'.differentiable_on
   hf''.differentiable_on (λ x _, hf''_nonpos x)
 
diff --git a/src/analysis/convex/exposed.lean b/src/analysis/convex/exposed.lean
index d1271b51af267..41cb5c62180ca 100644
--- a/src/analysis/convex/exposed.lean
+++ b/src/analysis/convex/exposed.lean
@@ -164,12 +164,12 @@ begin
   exact hC.inter_left hCA,
 end
 
-protected lemma is_extreme [normed_space ℝ E]  (hAB : is_exposed ℝ A B) :
-  is_extreme ℝ A B :=
+protected lemma is_extreme (hAB : is_exposed 𝕜 A B) :
+  is_extreme 𝕜 A B :=
 begin
   refine ⟨hAB.subset, λ x₁ x₂ hx₁A hx₂A x hxB hx, _⟩,
   obtain ⟨l, rfl⟩ := hAB ⟨x, hxB⟩,
-  have hl : convex_on univ l := l.to_linear_map.convex_on convex_univ,
+  have hl : convex_on 𝕜 univ l := l.to_linear_map.convex_on convex_univ,
   have hlx₁ := hxB.2 x₁ hx₁A,
   have hlx₂ := hxB.2 x₂ hx₂A,
   refine ⟨⟨hx₁A, λ y hy, _⟩, ⟨hx₂A, λ y hy, _⟩⟩,
@@ -179,8 +179,8 @@ begin
     exact hxB.2 y hy }
 end
 
-protected lemma convex [normed_space ℝ E] (hAB : is_exposed ℝ A B) (hA : convex ℝ A) :
-  convex ℝ B :=
+protected lemma convex (hAB : is_exposed 𝕜 A B) (hA : convex 𝕜 A) :
+  convex 𝕜 B :=
 begin
   obtain rfl | hB := B.eq_empty_or_nonempty,
   { exact convex_empty },
diff --git a/src/analysis/convex/extrema.lean b/src/analysis/convex/extrema.lean
index 3287557a73a8c..a4da82694ffb6 100644
--- a/src/analysis/convex/extrema.lean
+++ b/src/analysis/convex/extrema.lean
@@ -26,7 +26,7 @@ open_locale classical
 Helper lemma for the more general case: `is_min_on.of_is_local_min_on_of_convex_on`.
 -/
 lemma is_min_on.of_is_local_min_on_of_convex_on_Icc {f : ℝ → β} {a b : ℝ} (a_lt_b : a < b)
-  (h_local_min : is_local_min_on f (Icc a b) a) (h_conv : convex_on (Icc a b) f) :
+  (h_local_min : is_local_min_on f (Icc a b) a) (h_conv : convex_on ℝ (Icc a b) f) :
   ∀ x ∈ Icc a b, f a ≤ f x :=
 begin
   by_contradiction H_cont,
@@ -60,7 +60,7 @@ end
 A local minimum of a convex function is a global minimum, restricted to a set `s`.
 -/
 lemma is_min_on.of_is_local_min_on_of_convex_on {f : E → β} {a : E}
-  (a_in_s : a ∈ s) (h_localmin : is_local_min_on f s a) (h_conv : convex_on s f) :
+  (a_in_s : a ∈ s) (h_localmin : is_local_min_on f s a) (h_conv : convex_on ℝ s f) :
   ∀ x ∈ s, f a ≤ f x :=
 begin
   by_contradiction H_cont,
@@ -91,18 +91,18 @@ end
 
 /-- A local maximum of a concave function is a global maximum, restricted to a set `s`. -/
 lemma is_max_on.of_is_local_max_on_of_concave_on {f : E → β} {a : E}
-  (a_in_s : a ∈ s) (h_localmax: is_local_max_on f s a) (h_conc : concave_on s f) :
+  (a_in_s : a ∈ s) (h_localmax: is_local_max_on f s a) (h_conc : concave_on ℝ s f) :
   ∀ x ∈ s, f x ≤ f a :=
 @is_min_on.of_is_local_min_on_of_convex_on
   _ (order_dual β) _ _ _ _ _ _ _ _ s f a a_in_s h_localmax h_conc
 
 /-- A local minimum of a convex function is a global minimum. -/
 lemma is_min_on.of_is_local_min_of_convex_univ {f : E → β} {a : E}
-  (h_local_min : is_local_min f a) (h_conv : convex_on univ f) : ∀ x, f a ≤ f x :=
+  (h_local_min : is_local_min f a) (h_conv : convex_on ℝ univ f) : ∀ x, f a ≤ f x :=
 λ x, (is_min_on.of_is_local_min_on_of_convex_on (mem_univ a)
         (is_local_min.on h_local_min univ) h_conv) x (mem_univ x)
 
 /-- A local maximum of a concave function is a global maximum. -/
 lemma is_max_on.of_is_local_max_of_convex_univ {f : E → β} {a : E}
-  (h_local_max : is_local_max f a) (h_conc : concave_on univ f) : ∀ x, f x ≤ f a :=
+  (h_local_max : is_local_max f a) (h_conc : concave_on ℝ univ f) : ∀ x, f x ≤ f a :=
 @is_min_on.of_is_local_min_of_convex_univ _ (order_dual β) _ _ _ _ _ _ _ _ f a h_local_max h_conc
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index fb521e6d13845..0ad50dcf9c3e9 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -13,13 +13,13 @@ inequality. The integral version can be found in `analysis.convex.integral`.
 
 A function `f : E → β` is `convex_on` a set `s` if `s` is itself a convex set, and for any two
 points `x y ∈ s`, the segment joining `(x, f x)` to `(y, f y)` is above the graph of `f`.
-Equivalently, `convex_on 𝕜 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f z.1 ≤ p.2}` is
+Equivalently, `convex_on 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f z.1 ≤ p.2}` is
 a convex set.
 
 ## Main declarations
 
-* `convex_on 𝕜 𝕜 s f`: The function `f` is convex on `s` with scalars `𝕜`.
-* `concave_on 𝕜 𝕜 s f`: The function `f` is concave on `s` with scalars `𝕜`.
+* `convex_on 𝕜 s f`: The function `f` is convex on `s` with scalars `𝕜`.
+* `concave_on 𝕜 s f`: The function `f` is concave on `s` with scalars `𝕜`.
 * `convex_on.map_linear_combination_le` `convex_on.map_sum_le`: Convex Jensen's inequality.
 -/
 
@@ -479,8 +479,8 @@ begin
 end⟩
 
 lemma concave_on_iff_div {f : E → β} :
-  concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → 0 < a + b
-  → (a/(a+b)) • f x + (b/(a+b)) • f y ≤ f ((a/(a+b)) • x + (b/(a+b)) • y) :=
+  concave_on 𝕜 s f ↔ convex 𝕜 s ∧ ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b
+  → 0 < a + b → (a/(a+b)) • f x + (b/(a+b)) • f y ≤ f ((a/(a+b)) • x + (b/(a+b)) • y) :=
 @convex_on_iff_div _ _ (order_dual β) _ _ _ _ _ _ _
 
 /-- For a function `f` defined on a convex subset `D` of `𝕜`, if for any three points `x < y < z`
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index 0bee342349f07..ccd9551af1c35 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -108,7 +108,7 @@ to `s`, then the value of `g` at the average value of `f` is less than or equal
 of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
 `convex.map_linear_combination_le` for a finite sum version of this lemma. -/
 lemma convex_on.map_smul_integral_le [is_finite_measure μ] {s : set E} {g : E → ℝ}
-  (hg : convex_on s g) (hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) {f : α → E}
+  (hg : convex_on ℝ s g) (hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) {f : α → E}
   (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
   g ((μ univ).to_real⁻¹ • ∫ x, f x ∂μ) ≤ (μ univ).to_real⁻¹ • ∫ x, g (f x) ∂μ :=
 begin
@@ -128,7 +128,7 @@ end
 of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also `convex.map_sum_le` for a
 finite sum version of this lemma. -/
 lemma convex_on.map_integral_le [is_probability_measure μ] {s : set E} {g : E → ℝ}
-  (hg : convex_on s g) (hgc : continuous_on g s) (hsc : is_closed s) {f : α → E}
+  (hg : convex_on ℝ s g) (hgc : continuous_on g s) (hsc : is_closed s) {f : α → E}
   (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
   g (∫ x, f x ∂μ) ≤ ∫ x, g (f x) ∂μ :=
 by simpa [measure_univ]
diff --git a/src/analysis/convex/specific_functions.lean b/src/analysis/convex/specific_functions.lean
index 93ff0be4c3c8e..3bca1de1375c8 100644
--- a/src/analysis/convex/specific_functions.lean
+++ b/src/analysis/convex/specific_functions.lean
@@ -16,7 +16,7 @@ In this file we prove that the following functions are convex:
   is convex on $(-∞, +∞)$;
 * `convex_on_pow` : for a natural $n$, the function $f(x)=x^n$ is convex on $[0, +∞)$;
 * `convex_on_fpow` : for an integer $m$, the function $f(x)=x^m$ is convex on $(0, +∞)$.
-* `convex_on_rpow : ∀ p : ℝ, 1 ≤ p → convex_on (Ici 0) (λ x, x ^ p)`
+* `convex_on_rpow : ∀ p : ℝ, 1 ≤ p → convex_on ℝ (Ici 0) (λ x, x ^ p)`
 * `concave_on_log_Ioi` and `concave_on_log_Iio`: log is concave on `Ioi 0` and `Iio 0` respectively.
 -/
 
@@ -24,12 +24,12 @@ open real set
 open_locale big_operators
 
 /-- `exp` is convex on the whole real line -/
-lemma convex_on_exp : convex_on univ exp :=
+lemma convex_on_exp : convex_on ℝ univ exp :=
 convex_on_univ_of_deriv2_nonneg differentiable_exp (by simp)
   (assume x, (iter_deriv_exp 2).symm ▸ le_of_lt (exp_pos x))
 
 /-- `x^n`, `n : ℕ` is convex on the whole real line whenever `n` is even -/
-lemma convex_on_pow_of_even {n : ℕ} (hn : even n) : convex_on set.univ (λ x : ℝ, x^n) :=
+lemma convex_on_pow_of_even {n : ℕ} (hn : even n) : convex_on ℝ set.univ (λ x : ℝ, x^n) :=
 begin
   apply convex_on_univ_of_deriv2_nonneg differentiable_pow,
   { simp only [deriv_pow', differentiable.mul, differentiable_const, differentiable_pow] },
@@ -40,7 +40,7 @@ begin
 end
 
 /-- `x^n`, `n : ℕ` is convex on `[0, +∞)` for all `n` -/
-lemma convex_on_pow (n : ℕ) : convex_on (Ici 0) (λ x : ℝ, x^n) :=
+lemma convex_on_pow (n : ℕ) : convex_on ℝ (Ici 0) (λ x : ℝ, x^n) :=
 begin
   apply convex_on_of_deriv2_nonneg (convex_Ici _) (continuous_pow n).continuous_on
     differentiable_on_pow,
@@ -76,7 +76,7 @@ begin
 end
 
 /-- `x^m`, `m : ℤ` is convex on `(0, +∞)` for all `m` -/
-lemma convex_on_fpow (m : ℤ) : convex_on (Ioi 0) (λ x : ℝ, x^m) :=
+lemma convex_on_fpow (m : ℤ) : convex_on ℝ (Ioi 0) (λ x : ℝ, x^m) :=
 begin
   have : ∀ n : ℤ, differentiable_on ℝ (λ x, x ^ n) (Ioi (0 : ℝ)),
     from λ n, differentiable_on_fpow _ _ (or.inl $ lt_irrefl _),
@@ -91,7 +91,7 @@ begin
     exact int_prod_range_nonneg _ _ (nat.even_bit0 1) }
 end
 
-lemma convex_on_rpow {p : ℝ} (hp : 1 ≤ p) : convex_on (Ici 0) (λ x : ℝ, x^p) :=
+lemma convex_on_rpow {p : ℝ} (hp : 1 ≤ p) : convex_on ℝ (Ici 0) (λ x : ℝ, x^p) :=
 begin
   have A : deriv (λ (x : ℝ), x ^ p) = λ x, p * x^(p-1), by { ext x, simp [hp] },
   apply convex_on_of_deriv2_nonneg (convex_Ici 0),
@@ -108,7 +108,7 @@ begin
     exact mul_nonneg (sub_nonneg_of_le hp) (rpow_nonneg_of_nonneg (le_of_lt hx) _) }
 end
 
-lemma concave_on_log_Ioi : concave_on (Ioi 0) log :=
+lemma concave_on_log_Ioi : concave_on ℝ (Ioi 0) log :=
 begin
   have h₁ : Ioi 0 ⊆ ({0} : set ℝ)ᶜ,
   { intros x hx hx',
@@ -126,7 +126,7 @@ begin
     exact neg_nonpos.mpr (inv_nonneg.mpr (sq_nonneg x)) }
 end
 
-lemma concave_on_log_Iio : concave_on (Iio 0) log :=
+lemma concave_on_log_Iio : concave_on ℝ (Iio 0) log :=
 begin
   have h₁ : Iio 0 ⊆ ({0} : set ℝ)ᶜ,
   { intros x hx hx',
diff --git a/src/analysis/convex/topology.lean b/src/analysis/convex/topology.lean
index 6d5a0a8559cf1..504e686849074 100644
--- a/src/analysis/convex/topology.lean
+++ b/src/analysis/convex/topology.lean
@@ -181,7 +181,7 @@ section normed_space
 variables [normed_group E] [normed_space ℝ E]
 
 lemma convex_on_dist (z : E) (s : set E) (hs : convex ℝ s) :
-  convex_on s (λz', dist z' z) :=
+  convex_on ℝ s (λz', dist z' z) :=
 and.intro hs $
 assume x y hx hy a b ha hb hab,
 calc

From 452f3d89ea0b3e5ba08f5a61850b2ef641fffc72 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Thu, 23 Sep 2021 23:18:30 +0200
Subject: [PATCH 12/17] fix mean_inequalities

---
 docs/overview.yaml                  |  2 +-
 src/analysis/convex/integral.lean   | 12 ++++++------
 src/analysis/mean_inequalities.lean | 14 +++++++-------
 3 files changed, 14 insertions(+), 14 deletions(-)

diff --git a/docs/overview.yaml b/docs/overview.yaml
index e3f3c0676c2f1..bcadbe8ba9817 100644
--- a/docs/overview.yaml
+++ b/docs/overview.yaml
@@ -272,7 +272,7 @@ Analysis:
   Convexity:
     convex function: 'convex_on'
     characterization of convexity: 'convex_on_of_deriv2_nonneg'
-    Jensen's inequality (finite sum version): 'convex_on.map_sum_le'
+    Jensen's inequality (finite sum version): 'convex_on.map_linear_combination_le'
     Jensen's inequality (integral version): 'convex_on.map_integral_le'
     convexity inequalities: 'analysis/mean_inequalities.html'
     Carathéodory's theorem: 'convex_hull_eq_union'
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index ccd9551af1c35..f75d8e751ac72 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -32,7 +32,7 @@ In this file we prove four theorems:
   continuous on a convex closed set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a
   function sending `μ`-a.e. points to `s`, then the value of `g` at the expected value of `f` is
   less than or equal to the expected value of `g ∘ f` provided that both `f` and `g ∘ f` are
-  integrable. See also `convex.map_sum_le` for a finite sum version of this lemma.
+  integrable. See also `convex_on.map_linear_combination_le` for a finite sum version of this lemma.
 
 ## Tags
 
@@ -122,11 +122,11 @@ begin
     using (ht_conv.smul_integral_mem ht_closed hμ ht_mem (hfi.prod_mk hgi)).2
 end
 
-/-- Jensen's inequality: if a function `g : E → ℝ` is convex and continuous on a convex closed set
-`s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e. points to
-`s`, then the value of `g` at the expected value of `f` is less than or equal to the expected value
-of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also `convex.map_sum_le` for a
-finite sum version of this lemma. -/
+/-- Convex **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex
+closed set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e.
+points to `s`, then the value of `g` at the expected value of `f` is less than or equal to the
+expected value of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
+`convex_on.map_linear_combination_le` for a finite sum version of this lemma. -/
 lemma convex_on.map_integral_le [is_probability_measure μ] {s : set E} {g : E → ℝ}
   (hg : convex_on ℝ s g) (hgc : continuous_on g s) (hsc : is_closed s) {f : α → E}
   (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
diff --git a/src/analysis/mean_inequalities.lean b/src/analysis/mean_inequalities.lean
index 6a721629e4ca7..0ac50272fc803 100644
--- a/src/analysis/mean_inequalities.lean
+++ b/src/analysis/mean_inequalities.lean
@@ -15,7 +15,7 @@ these inequalities are available in `measure_theory.mean_inequalities`.
 
 ## Main theorems
 
-### AM-GM inequality:
+### AM-GM inequality
 
 The inequality says that the geometric mean of a tuple of non-negative numbers is less than or equal
 to their arithmetic mean. We prove the weighted version of this inequality: if $w$ and $z$
@@ -127,8 +127,8 @@ begin
   -- If all numbers `z i` with non-zero weight are positive, then we apply Jensen's inequality
   -- for `exp` and numbers `log (z i)` with weights `w i`.
   { simp only [not_exists, not_and, ne.def, not_not] at A,
-    have := convex_on_exp.map_sum_le hw hw' (λ i _, set.mem_univ $ log (z i)),
-    simp only [exp_sum, (∘), smul_eq_mul, mul_comm (w _) (log _)] at this,
+    have := convex_on_exp.map_linear_combination_le hw hw' (λ i _, set.mem_univ $ log (z i)),
+    simp only [linear_combination, exp_sum, (∘), smul_eq_mul, mul_comm (w _) (log _)] at this,
     convert this using 1; [apply prod_congr rfl, apply sum_congr rfl]; intros i hi,
     { cases eq_or_lt_of_le (hz i hi) with hz hz,
       { simp [A i hi hz.symm] },
@@ -141,22 +141,22 @@ end
 theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
   (hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (n : ℕ) :
   (∑ i in s, w i * z i) ^ n ≤ ∑ i in s, (w i * z i ^ n) :=
-(convex_on_pow n).map_sum_le hw hw' hz
+(convex_on_pow n).map_linear_combination_le hw hw' hz
 
 theorem pow_arith_mean_le_arith_mean_pow_of_even (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
   (hw' : ∑ i in s, w i = 1) {n : ℕ} (hn : even n) :
   (∑ i in s, w i * z i) ^ n ≤ ∑ i in s, (w i * z i ^ n) :=
-(convex_on_pow_of_even hn).map_sum_le hw hw' (λ _ _, trivial)
+(convex_on_pow_of_even hn).map_linear_combination_le hw hw' (λ _ _, trivial)
 
 theorem fpow_arith_mean_le_arith_mean_fpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
   (hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ) :
   (∑ i in s, w i * z i) ^ m ≤ ∑ i in s, (w i * z i ^ m) :=
-(convex_on_fpow m).map_sum_le hw hw' hz
+(convex_on_fpow m).map_linear_combination_le hw hw' hz
 
 theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
   (hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
   (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, (w i * z i ^ p) :=
-(convex_on_rpow hp).map_sum_le hw hw' hz
+(convex_on_rpow hp).map_linear_combination_le hw hw' hz
 
 theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
   (hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :

From 113551619c52013017960984d43b324a4f4da74c Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Fri, 24 Sep 2021 22:32:22 +0200
Subject: [PATCH 13/17] revert some more

---
 docs/overview.yaml                |  2 +-
 src/analysis/convex/basic.lean    |  4 ++--
 src/analysis/convex/function.lean |  2 +-
 src/analysis/convex/integral.lean | 23 +++++++++++------------
 4 files changed, 15 insertions(+), 16 deletions(-)

diff --git a/docs/overview.yaml b/docs/overview.yaml
index 516101bd4fac9..e3f3c0676c2f1 100644
--- a/docs/overview.yaml
+++ b/docs/overview.yaml
@@ -272,7 +272,7 @@ Analysis:
   Convexity:
     convex function: 'convex_on'
     characterization of convexity: 'convex_on_of_deriv2_nonneg'
-    Jensen's inequality (finite sum version): 'convex_on.map_center_mass_le'
+    Jensen's inequality (finite sum version): 'convex_on.map_sum_le'
     Jensen's inequality (integral version): 'convex_on.map_integral_le'
     convexity inequalities: 'analysis/mean_inequalities.html'
     Carathéodory's theorem: 'convex_hull_eq_union'
diff --git a/src/analysis/convex/basic.lean b/src/analysis/convex/basic.lean
index ec32d6f6fbcc8..7438232083a1e 100644
--- a/src/analysis/convex/basic.lean
+++ b/src/analysis/convex/basic.lean
@@ -404,7 +404,7 @@ section add_comm_monoid
 variables [add_comm_monoid E]
 
 /-- Convexity of sets. -/
-def convex [has_scalar 𝕜 E](s : set E) :=
+def convex [has_scalar 𝕜 E] (s : set E) :=
 ∀ ⦃x y : E⦄, x ∈ s → y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 →
   a • x + b • y ∈ s
 
@@ -1035,7 +1035,7 @@ end convex_hull
 
 section simplex
 
-variables (ι : Type*) [ordered_semiring 𝕜] [add_comm_group E] [fintype ι] [module 𝕜 E]
+variables (ι : Type*) [ordered_semiring 𝕜] [fintype ι]
 
 /-- The standard simplex in the space of functions `ι → 𝕜` is the set of vectors with non-negative
 coordinates with total sum `1`. This is the free object in the category of convex spaces.-/
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index 81cbf0d8bc61c..7a39912be1a38 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -15,7 +15,7 @@ inequality. The integral version can be found in `analysis.convex.integral`.
 
 A function `f : E → β` is `convex_on` a set `s` if `s` is itself a convex set, and for any two
 points `x y ∈ s`, the segment joining `(x, f x)` to `(y, f y)` is above the graph of `f`.
-Equivalently, `convex_on 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f z.1 ≤ p.2}` is
+Equivalently, `convex_on 𝕜 f s` means that the epigraph `{p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2}` is
 a convex set.
 
 ## Main declarations
diff --git a/src/analysis/convex/integral.lean b/src/analysis/convex/integral.lean
index 35556debff1ca..cabfade17b69c 100644
--- a/src/analysis/convex/integral.lean
+++ b/src/analysis/convex/integral.lean
@@ -18,21 +18,20 @@ In this file we prove four theorems:
 
 * `convex.integral_mem`: if `μ` is a probability measure on `α`, `s` is a convex closed set in `E`,
   and `f` is an integrable function sending `μ`-a.e. points to `s`, then the expected value of `f`
-  belongs to `s`: `∫ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum
-  version of this lemma.
+  belongs to `s`: `∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this
+  lemma.
 
 * `convex_on.map_smul_integral_le`: Convex Jensen's inequality: If a function `g : E → ℝ` is convex
   and continuous on a convex closed set `s`, `μ` is a finite non-zero measure on `α`, and
   `f : α → E` is a function sending `μ`-a.e. points to `s`, then the value of `g` at the average
   value of `f` is less than or equal to the average value of `g ∘ f` provided that both `f` and
-  `g ∘ f` are integrable. See also `convex_on.map_center_mass_le` for a finite sum version of
-  this lemma.
+  `g ∘ f` are integrable. See also `convex_on.map_sum_le` for a finite sum version of this lemma.
 
 * `convex_on.map_integral_le`: Convex Jensen's inequality: If a function `g : E → ℝ` is convex and
   continuous on a convex closed set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a
   function sending `μ`-a.e. points to `s`, then the value of `g` at the expected value of `f` is
   less than or equal to the expected value of `g ∘ f` provided that both `f` and `g ∘ f` are
-  integrable. See also `convex_on.map_center_mass_le` for a finite sum version of this lemma.
+  integrable. See also `convex_on.map_sum_le` for a finite sum version of this lemma.
 
 ## Tags
 
@@ -67,9 +66,9 @@ begin
     by rw [← (F n).sum_range_measure_preimage_singleton, @ennreal.to_real_sum _ _
       (λ y, μ ((F n) ⁻¹' {y})) (λ _ _, (measure_ne_top _ _))],
   rw [← this, simple_func.integral],
-  refine hs.center_mass_mem' (λ _ _, ennreal.to_real_nonneg) _ _,
-  { rw [this, ne.def, ennreal.to_real_eq_zero_iff, measure.measure_univ_eq_zero],
-    exact not_or hμ (measure_ne_top _ _) },
+  refine hs.center_mass_mem (λ _ _, ennreal.to_real_nonneg) _ _,
+  { rw [this, ennreal.to_real_pos_iff, pos_iff_ne_zero, ne.def, measure.measure_univ_eq_zero],
+    exact ⟨hμ, measure_ne_top _ _⟩ },
   { simp only [simple_func.mem_range],
     rintros _ ⟨x, rfl⟩,
     exact simple_func.approx_on_mem hfm h₀ n x }
@@ -77,8 +76,8 @@ end
 
 /-- If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
 integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
-`(μ univ).to_real⁻¹ • ∫ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum
-version of this lemma. -/
+`(μ univ).to_real⁻¹ • ∫ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version
+of this lemma. -/
 lemma convex.smul_integral_mem
   [is_finite_measure μ] {s : set E} (hs : convex ℝ s) (hsc : is_closed s)
   (hμ : μ ≠ 0) {f : α → E} (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) :
@@ -105,8 +104,8 @@ by simpa [measure_univ] using hs.smul_integral_mem hsc (is_probability_measure.n
 /-- Jensen's inequality: if a function `g : E → ℝ` is convex and continuous on a convex closed set
 `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending `μ`-a.e. points
 to `s`, then the value of `g` at the average value of `f` is less than or equal to the average value
-of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also
-`convex.map_center_mass_le` for a finite sum version of this lemma. -/
+of `g ∘ f` provided that both `f` and `g ∘ f` are integrable. See also `convex.map_center_mass_le`
+for a finite sum version of this lemma. -/
 lemma convex_on.map_smul_integral_le [is_finite_measure μ] {s : set E} {g : E → ℝ}
   (hg : convex_on ℝ s g) (hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) {f : α → E}
   (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :

From 6d179d2cd77158cb72db0d3a853126d3898e2b7a Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Sat, 25 Sep 2021 08:53:26 +0200
Subject: [PATCH 14/17] fix combination

---
 src/analysis/convex/combination.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 03c49245371b2..041d1e23be513 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -278,7 +278,7 @@ The map is defined in terms of operations on `(s → ℝ) →ₗ[ℝ] ℝ` so th
 to prove that this map is linear. -/
 lemma set.finite.convex_hull_eq_image {s : set E} (hs : finite s) :
   convex_hull R s = by haveI := hs.fintype; exact
-    (⇑(∑ x : s, (@linear_map.proj R s _ (λ i, R) _ _ x).smul_right x.1)) '' (std_simplex s R) :=
+    (⇑(∑ x : s, (@linear_map.proj R s _ (λ i, R) _ _ x).smul_right x.1)) '' (std_simplex R s) :=
 begin
   rw [← convex_hull_basis_eq_std_simplex, ← linear_map.convex_hull_image, ← set.range_comp, (∘)],
   apply congr_arg,

From ed6fc38ac08dcc2bd7dc46109571670b4930128b Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Sat, 25 Sep 2021 14:20:45 +0200
Subject: [PATCH 15/17] fix Jensen and maximum principle

---
 src/analysis/convex/combination.lean |  2 +-
 src/analysis/convex/exposed.lean     |  2 +-
 src/analysis/convex/function.lean    | 78 +++++++++++++++-------------
 3 files changed, 45 insertions(+), 37 deletions(-)

diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 041d1e23be513..63cf784c230e5 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -27,7 +27,7 @@ open_locale big_operators classical
 
 universes u u'
 variables {R E F ι ι' : Type*}
-  [linear_ordered_field R] [add_comm_group E] [module R E] [add_comm_group F] [module R F]
+  [linear_ordered_field R] [add_comm_monoid E] [module R E] [add_comm_group F] [module R F]
   {s : set E}
 
 /-- Center of mass of a finite collection of points with prescribed weights.
diff --git a/src/analysis/convex/exposed.lean b/src/analysis/convex/exposed.lean
index 41cb5c62180ca..f12467c6cb686 100644
--- a/src/analysis/convex/exposed.lean
+++ b/src/analysis/convex/exposed.lean
@@ -186,7 +186,7 @@ begin
   { exact convex_empty },
   obtain ⟨l, rfl⟩ := hAB hB,
   exact λ x₁ x₂ hx₁ hx₂ a b ha hb hab, ⟨hA hx₁.1 hx₂.1 ha hb hab, λ y hy,
-    ((l.to_linear_map.concave_on convex_univ).concave_le _
+    ((l.to_linear_map.concave_on convex_univ).concave_ge _
     ⟨mem_univ _, hx₁.2 y hy⟩ ⟨mem_univ _, hx₂.2 y hy⟩ ha hb hab).2⟩,
 end
 
diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index 7a39912be1a38..527fe569a1d90 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -108,7 +108,7 @@ lemma convex_on.convex_le (hf : convex_on 𝕜 s f) (r : β) :
                                                   (smul_le_smul_of_nonneg hy.2 hb)
                   ... = r                     : convex.combo_self hab r⟩
 
-lemma concave_on.concave_le (hf : concave_on 𝕜 s f) (r : β) :
+lemma concave_on.concave_ge (hf : concave_on 𝕜 s f) (r : β) :
   convex 𝕜 {x ∈ s | r ≤ f x} :=
 @convex_on.convex_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ f hf r
 
@@ -607,8 +607,8 @@ variables [linear_ordered_field 𝕜] [add_comm_monoid E] [ordered_add_comm_mono
 
 /-- Convex **Jensen's inequality**, `finset.center_mass` version. -/
 lemma convex_on.map_center_mass_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
-  (h₁ : ∑ i in t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
-  f (t.center_mass p w) ≤ t.center_mass (f ∘ p) w :=
+  (h₁ : 0 < ∑ i in t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
+  f (t.center_mass w p) ≤ t.center_mass w (f ∘ p) :=
 begin
   have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2},
     from λ i hi, ⟨hmem i hi, le_rfl⟩,
@@ -619,25 +619,21 @@ end
 
 /-- Concave **Jensen's inequality**, `finset.center_mass` version. -/
 lemma concave_on.le_map_center_mass (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
-  (h₁ : ∑ i in t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) :
-  t.center_mass (f ∘ p) w ≤ f (t.center_mass p w) :=
+  (h₁ : 0 < ∑ i in t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
+  t.center_mass w (f ∘ p) ≤ f (t.center_mass w p) :=
 @convex_on.map_center_mass_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ _ _ _ _ hf h₀ h₁ hmem
 
 /-- Convex **Jensen's inequality**, `finset.sum` version. -/
-lemma convex_on.map_sum_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i ≠ 0)
+lemma convex_on.map_sum_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
   (hmem : ∀ i ∈ t, p i ∈ s) :
-  f ((∑ i in t, w i)⁻¹ • ∑ i in t, w i • p i) ≤ (∑ i in t, w i)⁻¹ • ∑ i in t, w i • f (p i) :=
-begin
-  rw [←center_mass, ←center_mass_smul_right, ←center_mass,
-    ←center_mass_smul_right],
-  exact hf.map_center_mass_le (center_mass_normalize_weight_nonneg h₀)
-    (center_mass_normalize_weight_sum h₁) hmem,
-end
+  f (∑ i in t, w i • p i) ≤ ∑ i in t, w i • f (p i) :=
+by simpa only [center_mass, h₁, inv_one, one_smul]
+  using hf.map_center_mass_le h₀ (h₁.symm ▸ zero_lt_one) hmem
 
 /-- Concave **Jensen's inequality**, `finset.sum` version. -/
-lemma concave_on.le_map_sum (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i ≠ 0)
+lemma concave_on.le_map_sum (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
   (hmem : ∀ i ∈ t, p i ∈ s) :
-  (∑ i in t, w i)⁻¹ • ∑ i in t, w i • f (p i) ≤ f ((∑ i in t, w i)⁻¹ • ∑ i in t, w i • p i) :=
+  ∑ i in t, w i • f (p i) ≤ f (∑ i in t, w i • p i) :=
 @convex_on.map_sum_le 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ _ _ _ _ hf h₀ h₁ hmem
 
 end jensen
@@ -645,40 +641,52 @@ end jensen
 /-! ### Maximum principle -/
 
 section maximum_principle
-variables [linear_ordered_field 𝕜] [add_comm_monoid E] [ordered_add_comm_monoid β] [module 𝕜 E]
-  [mul_action_with_zero 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f : E → β} {t : finset ι} {w : ι → 𝕜}
+variables [linear_ordered_field 𝕜] [add_comm_monoid E] [linear_ordered_add_comm_group β]
+  [module 𝕜 E] [module 𝕜 β] [ordered_smul 𝕜 β] {s : set E} {f : E → β} {t : finset ι} {w : ι → 𝕜}
   {p : ι → E}
 
-/-- If a function `f` is convex on `s` takes value `y` at the center of mass of some points
-`p i ∈ s`, then for some `i` we have `y ≤ f (p i)`. -/
-lemma convex_on.exists_ge_of_center_mass {f : E → 𝕜} (h : convex_on 𝕜 s f)
-  (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i in t, w i = 1) (hz : ∀ i ∈ t, p i ∈ s) :
-  ∃ i ∈ t, f (t.center_mass p w) ≤ f (p i) :=
+/-- If a function `f` is convex on `s`, then the value it takes at some center of mass of points of
+`s` is less than the value it takes on one of those points. -/
+lemma convex_on.exists_ge_of_center_mass (h : convex_on 𝕜 s f)
+  (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : 0 < ∑ i in t, w i) (hp : ∀ i ∈ t, p i ∈ s) :
+  ∃ i ∈ t, f (t.center_mass w p) ≤ f (p i) :=
 begin
-  set y := t.center_mass p w,
+  set y := t.center_mass w p,
   suffices h : ∃ i ∈ t.filter (λ i, w i ≠ 0), w i • f y ≤ w i • (f ∘ p) i,
   { obtain ⟨i, hi, hfi⟩ := h,
     rw mem_filter at hi,
     exact ⟨i, hi.1, (smul_le_smul_iff_of_pos $ (hw₀ i hi.1).lt_of_ne hi.2.symm).1 hfi⟩ },
-  refine exists_le_of_sum_le _ _,
-  { have := @zero_ne_one 𝕜 _ _,
-    rw [←hw₁, ←sum_filter_ne_zero] at this,
-    exact nonempty_of_sum_ne_zero this.symm },
-  { rw [←center_mass, ←center_mass, finset.center_mass_filter_ne_zero,
-      finset.center_mass_filter_ne_zero, center_mass_const_left, hw₁, one_smul],
-    exact h.map_center_mass_le hw₀ hw₁ hz }
+  have hw' : (0 : 𝕜) < ∑ i in filter (λ i, w i ≠ 0) t, w i := by rwa sum_filter_ne_zero,
+  refine exists_le_of_sum_le (nonempty_of_sum_ne_zero hw'.ne') _,
+  rw [←sum_smul, ←smul_le_smul_iff_of_pos (inv_pos.2 hw'), inv_smul_smul' hw'.ne',
+    ←finset.center_mass, finset.center_mass_filter_ne_zero],
+  exact h.map_center_mass_le hw₀ hw₁ hp,
+  apply_instance,
 end
 
+/-- If a function `f` is concave on `s`, then the value it takes at some center of mass of points of
+`s` is greater than the value it takes on one of those points. -/
+lemma concave_on.exists_le_of_center_mass (h : concave_on 𝕜 s f)
+  (hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : 0 < ∑ i in t, w i) (hp : ∀ i ∈ t, p i ∈ s) :
+  ∃ i ∈ t, f (p i) ≤ f (t.center_mass w p) :=
+@convex_on.exists_ge_of_center_mass 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ _ _ _ _ h hw₀ hw₁ hp
+
 /-- Maximum principle for convex functions. If a function `f` is convex on the convex hull of `s`,
-then `f` reaches a maximum on `convex_hull 𝕜 s` outside of `s`. -/
-lemma convex_on.exists_ge_of_mem_convex_hull {f : E → 𝕜} (hf : convex_on 𝕜 (convex_hull 𝕜 s) f)
-  {x} (hx : x ∈ convex_hull 𝕜 s) : ∃ y ∈ s, f x ≤ f y :=
+then the eventual maximum of `f` on `convex_hull 𝕜 s` lies in `s`. -/
+lemma convex_on.exists_ge_of_mem_convex_hull (hf : convex_on 𝕜 (convex_hull 𝕜 s) f) {x}
+  (hx : x ∈ convex_hull 𝕜 s) : ∃ y ∈ s, f x ≤ f y :=
 begin
-  rw convex_hull_eq at hx,
+  rw _root_.convex_hull_eq at hx,
   obtain ⟨α, t, w, p, hw₀, hw₁, hp, rfl⟩ := hx,
-  rcases hf.exists_ge_of_center_mass hw₀ hw₁
+  rcases hf.exists_ge_of_center_mass hw₀ (hw₁.symm ▸ zero_lt_one)
     (λ i hi, subset_convex_hull 𝕜 s (hp i hi)) with ⟨i, hit, Hi⟩,
   exact ⟨p i, hp i hit, Hi⟩
 end
 
+/-- Minimum principle for concave functions. If a function `f` is concave on the convex hull of `s`,
+then the eventual minimum of `f` on `convex_hull 𝕜 s` lies in `s`. -/
+lemma concave_on.exists_le_of_mem_convex_hull (hf : concave_on 𝕜 (convex_hull 𝕜 s) f) {x}
+  (hx : x ∈ convex_hull 𝕜 s) : ∃ y ∈ s, f y ≤ f x :=
+@convex_on.exists_ge_of_mem_convex_hull 𝕜 E (order_dual β) _ _ _ _ _ _ _ _ hf _ hx
+
 end maximum_principle

From d84a2d1dff5fa3cf7cce014b64920adb17d306f2 Mon Sep 17 00:00:00 2001
From: YaelDillies <yael.dillies@gmail.com>
Date: Sun, 26 Sep 2021 10:54:12 +0200
Subject: [PATCH 16/17] revert std_simplex

---
 src/analysis/convex/basic.lean       | 17 ++++++++++-------
 src/analysis/convex/combination.lean | 15 +++++++--------
 src/analysis/convex/topology.lean    | 18 +++++++++---------
 3 files changed, 26 insertions(+), 24 deletions(-)

diff --git a/src/analysis/convex/basic.lean b/src/analysis/convex/basic.lean
index 7438232083a1e..caf06ae70bb4d 100644
--- a/src/analysis/convex/basic.lean
+++ b/src/analysis/convex/basic.lean
@@ -1035,18 +1035,21 @@ end convex_hull
 
 section simplex
 
-variables (ι : Type*) [ordered_semiring 𝕜] [fintype ι]
+variables (ι : Type*) [fintype ι]
 
-/-- The standard simplex in the space of functions `ι → 𝕜` is the set of vectors with non-negative
-coordinates with total sum `1`. This is the free object in the category of convex spaces.-/
-def std_simplex : set (ι → 𝕜) :=
+/-- The standard simplex in the space of functions `ι → ℝ` is the set
+of vectors with non-negative coordinates with total sum `1`. -/
+def std_simplex (ι : Type*) [fintype ι] (R : Type*) [ordered_semiring R] :
+  set (ι → R) :=
 {f | (∀ x, 0 ≤ f x) ∧ ∑ x, f x = 1}
 
+variables (R : Type*) [ordered_semiring R] {f : ι → R}
+
 lemma std_simplex_eq_inter :
-  std_simplex 𝕜 ι = (⋂ x, {f | 0 ≤ f x}) ∩ {f | ∑ x, f x = 1} :=
+  std_simplex ι R = (⋂ x, {f | 0 ≤ f x}) ∩ {f | ∑ x, f x = 1} :=
 by { ext f, simp only [std_simplex, set.mem_inter_eq, set.mem_Inter, set.mem_set_of_eq] }
 
-lemma convex_std_simplex : convex 𝕜 (std_simplex 𝕜 ι) :=
+lemma convex_std_simplex : convex R (std_simplex ι R) :=
 begin
   refine λ f g hf hg a b ha hb hab, ⟨λ x, _, _⟩,
   { apply_rules [add_nonneg, mul_nonneg, hf.1, hg.1] },
@@ -1057,7 +1060,7 @@ end
 
 variable {ι}
 
-lemma ite_eq_mem_std_simplex (i : ι) : (λ j, ite (i = j) (1:𝕜) 0) ∈ std_simplex 𝕜 ι :=
+lemma ite_eq_mem_std_simplex (i : ι) : (λ j, ite (i = j) (1:R) 0) ∈ std_simplex ι R :=
 ⟨λ j, by simp only; split_ifs; norm_num, by rw [finset.sum_ite_eq, if_pos (finset.mem_univ _)]⟩
 
 end simplex
diff --git a/src/analysis/convex/combination.lean b/src/analysis/convex/combination.lean
index 63cf784c230e5..8fa307c5dd118 100644
--- a/src/analysis/convex/combination.lean
+++ b/src/analysis/convex/combination.lean
@@ -26,9 +26,7 @@ open set
 open_locale big_operators classical
 
 universes u u'
-variables {R E F ι ι' : Type*}
-  [linear_ordered_field R] [add_comm_monoid E] [module R E] [add_comm_group F] [module R F]
-  {s : set E}
+variables {R E ι ι' : Type*} [linear_ordered_field R] [add_comm_monoid E] [module R E] {s : set E}
 
 /-- Center of mass of a finite collection of points with prescribed weights.
 Note that we require neither `0 ≤ w i` nor `∑ w = 1`. -/
@@ -256,11 +254,11 @@ end
 
 variables (ι) [fintype ι] {f : ι → R}
 
-/-- `std_simplex R ι` is the convex hull of the canonical basis in `ι → R`. -/
+/-- `std_simplex ι` is the convex hull of the canonical basis in `ι → ℝ`. -/
 lemma convex_hull_basis_eq_std_simplex :
-  convex_hull R (range $ λ(i j:ι), if i = j then (1:R) else 0) = std_simplex R ι :=
+  convex_hull R (range $ λ(i j:ι), if i = j then (1:R) else 0) = std_simplex ι R :=
 begin
-  refine subset.antisymm (convex_hull_min _ (convex_std_simplex R ι)) _,
+  refine subset.antisymm (convex_hull_min _ (convex_std_simplex ι R)) _,
   { rintros _ ⟨i, rfl⟩,
     exact ite_eq_mem_std_simplex R i },
   { rintros w ⟨hw₀, hw₁⟩,
@@ -273,12 +271,13 @@ variable {ι}
 
 /-- The convex hull of a finite set is the image of the standard simplex in `s → ℝ`
 under the linear map sending each function `w` to `∑ x in s, w x • x`.
+
 Since we have no sums over finite sets, we use sum over `@finset.univ _ hs.fintype`.
 The map is defined in terms of operations on `(s → ℝ) →ₗ[ℝ] ℝ` so that later we will not need
 to prove that this map is linear. -/
 lemma set.finite.convex_hull_eq_image {s : set E} (hs : finite s) :
   convex_hull R s = by haveI := hs.fintype; exact
-    (⇑(∑ x : s, (@linear_map.proj R s _ (λ i, R) _ _ x).smul_right x.1)) '' (std_simplex R s) :=
+    (⇑(∑ x : s, (@linear_map.proj R s _ (λ i, R) _ _ x).smul_right x.1)) '' (std_simplex s R) :=
 begin
   rw [← convex_hull_basis_eq_std_simplex, ← linear_map.convex_hull_image, ← set.range_comp, (∘)],
   apply congr_arg,
@@ -288,6 +287,6 @@ begin
 end
 
 /-- All values of a function `f ∈ std_simplex ι` belong to `[0, 1]`. -/
-lemma mem_Icc_of_mem_std_simplex (hf : f ∈ std_simplex R ι) (x) :
+lemma mem_Icc_of_mem_std_simplex (hf : f ∈ std_simplex ι R) (x) :
   f x ∈ Icc (0 : R) 1 :=
 ⟨hf.1 x, hf.2 ▸ finset.single_le_sum (λ y hy, hf.1 y) (finset.mem_univ x)⟩
diff --git a/src/analysis/convex/topology.lean b/src/analysis/convex/topology.lean
index 504e686849074..54adcb753725a 100644
--- a/src/analysis/convex/topology.lean
+++ b/src/analysis/convex/topology.lean
@@ -42,9 +42,9 @@ section std_simplex
 
 variables [fintype ι]
 
-/-- Every vector in `std_simplex ℝ ι` has `max`-norm at most `1`. -/
+/-- Every vector in `std_simplex ι` has `max`-norm at most `1`. -/
 lemma std_simplex_subset_closed_ball :
-  std_simplex ℝ ι ⊆ metric.closed_ball 0 1 :=
+  std_simplex ι ℝ ⊆ metric.closed_ball 0 1 :=
 begin
   assume f hf,
   rw [metric.mem_closed_ball, dist_zero_right],
@@ -56,18 +56,18 @@ end
 
 variable (ι)
 
-/-- `std_simplex ℝ ι` is bounded. -/
-lemma bounded_std_simplex : metric.bounded (std_simplex ℝ ι) :=
+/-- `std_simplex ι` is bounded. -/
+lemma bounded_std_simplex : metric.bounded (std_simplex ι ℝ) :=
 (metric.bounded_iff_subset_ball 0).2 ⟨1, std_simplex_subset_closed_ball⟩
 
-/-- `std_simplex ℝ ι` is closed. -/
-lemma is_closed_std_simplex : is_closed (std_simplex ℝ ι) :=
-(std_simplex_eq_inter ℝ ι).symm ▸ is_closed.inter
+/-- `std_simplex ι` is closed. -/
+lemma is_closed_std_simplex : is_closed (std_simplex ι ℝ) :=
+(std_simplex_eq_inter ι ℝ).symm ▸ is_closed.inter
   (is_closed_Inter $ λ i, is_closed_le continuous_const (continuous_apply i))
   (is_closed_eq (continuous_finset_sum _ $ λ x _, continuous_apply x) continuous_const)
 
-/-- `std_simplex ℝ ι` is compact. -/
-lemma compact_std_simplex : is_compact (std_simplex ℝ ι) :=
+/-- `std_simplex ι` is compact. -/
+lemma compact_std_simplex : is_compact (std_simplex ι ℝ) :=
 metric.compact_iff_closed_bounded.2 ⟨is_closed_std_simplex ι, bounded_std_simplex ι⟩
 
 end std_simplex

From 801d4944303f89a0319fd2382f124e0cdf0ddce5 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Ya=C3=ABl=20Dillies?= <yael.dillies@gmail.com>
Date: Mon, 27 Sep 2021 07:49:57 +0200
Subject: [PATCH 17/17] indentation

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
---
 src/analysis/convex/function.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/analysis/convex/function.lean b/src/analysis/convex/function.lean
index 527fe569a1d90..05fc99b565063 100644
--- a/src/analysis/convex/function.lean
+++ b/src/analysis/convex/function.lean
@@ -62,7 +62,7 @@ lemma convex_on.subset {f : E → β} {t : set E} (hf : convex_on 𝕜 t f) (hst
 ⟨hs, λ x y hx hy, hf.2 (hst hx) (hst hy)⟩
 
 lemma concave_on.subset {f : E → β} {t : set E} (hf : concave_on 𝕜 t f) (hst : s ⊆ t)
-(hs : convex 𝕜 s) : concave_on 𝕜 s f :=
+  (hs : convex 𝕜 s) : concave_on 𝕜 s f :=
 ⟨hs, λ x y hx hy, hf.2 (hst hx) (hst hy)⟩
 
 end has_scalar