From 1ffd04c075f81c09407a4be8dda6a21a232d30e2 Mon Sep 17 00:00:00 2001
From: mcdoll <moritz.doll@googlemail.com>
Date: Tue, 29 Mar 2022 09:36:08 +0000
Subject: [PATCH] feat(analysis/locally_convex): add balanced hull and core
 (#12537)

---
 .../locally_convex/balanced_core_hull.lean    | 246 ++++++++++++++++++
 1 file changed, 246 insertions(+)
 create mode 100644 src/analysis/locally_convex/balanced_core_hull.lean

diff --git a/src/analysis/locally_convex/balanced_core_hull.lean b/src/analysis/locally_convex/balanced_core_hull.lean
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+/-
+Copyright (c) 2022 Moritz Doll. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Doll
+-/
+import analysis.seminorm
+import order.closure
+
+/-!
+# Balanced Core and Balanced Hull
+
+## Main definitions
+
+* `balanced_core`: the largest balanced subset of a set `s`.
+* `balanced_hull`: the smallest balanced superset of a set `s`.
+
+## Main statements
+
+* `balanced_core_eq_Inter`: Characterization of the balanced core as an intersection over subsets.
+
+
+## Implementation details
+
+The balanced core and hull are implemented differently: for the core we take the obvious definition
+of the union over all balanced sets that are contained in `s`, whereas for the hull, we take the
+union over `r • s`, for `r` the scalars with `∥r∥ ≤ 1`. We show that `balanced_hull` has the
+defining properties of a hull in `balanced.hull_minimal` and `subset_balanced_hull`.
+For the core we need slightly stronger assumptions to obtain a characterization as an intersection,
+this is `balanced_core_eq_Inter`.
+
+## References
+
+* [Bourbaki, *Topological Vector Spaces*][bourbaki1987]
+
+## Tags
+
+balanced
+-/
+
+
+open set
+open_locale pointwise
+
+variables {𝕜 E ι : Type*}
+
+section balanced_hull
+
+section semi_normed_ring
+variables [semi_normed_ring 𝕜]
+
+section has_scalar
+variables [has_scalar 𝕜 E]
+
+variables (𝕜)
+
+/-- The largest balanced subset of `s`.-/
+def balanced_core (s : set E) := ⋃₀ {t : set E | balanced 𝕜 t ∧ t ⊆ s}
+
+/-- Helper definition to prove `balanced_core_eq_Inter`-/
+def balanced_core_aux (s : set E) := ⋂ (r : 𝕜) (hr : 1 ≤ ∥r∥), r • s
+
+/-- The smallest balanced superset of `s`.-/
+def balanced_hull (s : set E) := ⋃ (r : 𝕜) (hr : ∥r∥ ≤ 1), r • s
+
+variables {𝕜}
+
+lemma balanced_core_subset (s : set E) : balanced_core 𝕜 s ⊆ s :=
+begin
+  refine sUnion_subset (λ t ht, _),
+  simp only [mem_set_of_eq] at ht,
+  exact ht.2,
+end
+
+lemma balanced_core_mem_iff {s : set E} {x : E} : x ∈ balanced_core 𝕜 s ↔
+  ∃ t : set E, balanced 𝕜 t ∧ t ⊆ s ∧ x ∈ t :=
+by simp_rw [balanced_core, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc]
+
+lemma smul_balanced_core_subset (s : set E) {a : 𝕜} (ha : ∥a∥ ≤ 1) :
+  a • balanced_core 𝕜 s ⊆ balanced_core 𝕜 s :=
+begin
+  rw subset_def,
+  intros x hx,
+  rw mem_smul_set at hx,
+  rcases hx with ⟨y, hy, hx⟩,
+  rw balanced_core_mem_iff at hy,
+  rcases hy with ⟨t, ht1, ht2, hy⟩,
+  rw ←hx,
+  refine ⟨t, _, ht1 a ha (smul_mem_smul_set hy)⟩,
+  rw mem_set_of_eq,
+  exact ⟨ht1, ht2⟩,
+end
+
+lemma balanced_core_balanced (s : set E) : balanced 𝕜 (balanced_core 𝕜 s) :=
+λ _, smul_balanced_core_subset s
+
+/-- The balanced core of `t` is maximal in the sense that it contains any balanced subset
+`s` of `t`.-/
+lemma balanced.subset_core_of_subset {s t : set E} (hs : balanced 𝕜 s) (h : s ⊆ t):
+  s ⊆ balanced_core 𝕜 t :=
+begin
+  refine subset_sUnion_of_mem _,
+  rw [mem_set_of_eq],
+  exact ⟨hs, h⟩,
+end
+
+lemma balanced_core_aux_mem_iff (s : set E) (x : E) : x ∈ balanced_core_aux 𝕜 s ↔
+  ∀ (r : 𝕜) (hr : 1 ≤ ∥r∥), x ∈ r • s :=
+by rw [balanced_core_aux, set.mem_Inter₂]
+
+lemma balanced_hull_mem_iff (s : set E) (x : E) : x ∈ balanced_hull 𝕜 s ↔
+  ∃ (r : 𝕜) (hr : ∥r∥ ≤ 1), x ∈ r • s :=
+by rw [balanced_hull, set.mem_Union₂]
+
+/-- The balanced core of `s` is minimal in the sense that it is contained in any balanced superset
+`t` of `s`. -/
+lemma balanced.hull_subset_of_subset {s t : set E} (ht : balanced 𝕜 t) (h : s ⊆ t) :
+  balanced_hull 𝕜 s ⊆ t :=
+begin
+  intros x hx,
+  rcases (balanced_hull_mem_iff _ _).mp hx with ⟨r, hr, hx⟩,
+  rcases mem_smul_set.mp hx with ⟨y, hy, hx⟩,
+  rw ←hx,
+  exact balanced_mem ht (h hy) hr,
+end
+
+end has_scalar
+
+section add_comm_monoid
+
+variables [add_comm_monoid E] [module 𝕜 E]
+
+lemma balanced_core_nonempty_iff {s : set E} : (balanced_core 𝕜 s).nonempty ↔ (0 : E) ∈ s :=
+begin
+  split; intro h,
+  { cases h with x hx,
+    have h' : balanced 𝕜 (balanced_core 𝕜 s) := balanced_core_balanced s,
+    have h'' := h' 0 (has_le.le.trans norm_zero.le zero_le_one),
+    refine mem_of_subset_of_mem (subset.trans h'' (balanced_core_subset s)) _,
+    exact mem_smul_set.mpr ⟨x, hx, zero_smul _ _⟩ },
+  refine nonempty_of_mem (mem_of_subset_of_mem _ (mem_singleton 0)),
+  exact balanced.subset_core_of_subset zero_singleton_balanced (singleton_subset_iff.mpr h),
+end
+
+lemma balanced_core_zero_mem {s : set E} (hs: (0 : E) ∈ s) : (0 : E) ∈ balanced_core 𝕜 s :=
+balanced_core_mem_iff.mpr
+  ⟨{0}, zero_singleton_balanced, singleton_subset_iff.mpr hs, mem_singleton 0⟩
+
+variables (𝕜)
+
+lemma subset_balanced_hull [norm_one_class 𝕜] {s : set E} : s ⊆ balanced_hull 𝕜 s :=
+λ _ hx, (balanced_hull_mem_iff _ _).mpr ⟨1, norm_one.le, mem_smul_set.mp ⟨_, hx, one_smul _ _⟩⟩
+
+variables {𝕜}
+
+lemma balanced_hull.balanced (s : set E) : balanced 𝕜 (balanced_hull 𝕜 s) :=
+begin
+  intros a ha,
+  simp_rw [balanced_hull, smul_set_Union₂, subset_def, mem_Union₂],
+  intros x hx,
+  rcases hx with ⟨r, hr, hx⟩,
+  use [a • r],
+  split,
+  { rw smul_eq_mul,
+    refine has_le.le.trans (semi_normed_ring.norm_mul _ _) _,
+    refine mul_le_one ha (norm_nonneg r) hr },
+  rw smul_assoc,
+  exact hx,
+end
+
+end add_comm_monoid
+
+end semi_normed_ring
+
+section normed_field
+
+variables [normed_field 𝕜] [add_comm_group E] [module 𝕜 E]
+
+@[simp] lemma balanced_core_aux_empty : balanced_core_aux 𝕜 (∅ : set E) = ∅ :=
+begin
+  rw [balanced_core_aux, set.Inter₂_eq_empty_iff],
+  intros _,
+  simp only [smul_set_empty, mem_empty_eq, not_false_iff, exists_prop, and_true],
+  exact ⟨1, norm_one.ge⟩,
+end
+
+lemma balanced_core_aux_subset (s : set E) : balanced_core_aux 𝕜 s ⊆ s :=
+begin
+  rw subset_def,
+  intros x hx,
+  rw balanced_core_aux_mem_iff at hx,
+  have h := hx 1 norm_one.ge,
+  rw one_smul at h,
+  exact h,
+end
+
+lemma balanced_core_aux_balanced {s : set E} (h0 : (0 : E) ∈ balanced_core_aux 𝕜 s):
+  balanced 𝕜 (balanced_core_aux 𝕜 s) :=
+begin
+  intros a ha x hx,
+  rcases mem_smul_set.mp hx with ⟨y, hy, hx⟩,
+  by_cases (a = 0),
+  { simp[h] at hx,
+    rw ←hx,
+    exact h0 },
+  rw [←hx, balanced_core_aux_mem_iff],
+  rw balanced_core_aux_mem_iff at hy,
+  intros r hr,
+  have h'' : 1 ≤ ∥a⁻¹ • r∥ :=
+  begin
+    rw smul_eq_mul,
+    simp only [norm_mul, norm_inv],
+    exact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr,
+  end,
+  have h' := hy (a⁻¹ • r) h'',
+  rw smul_assoc at h',
+  exact (mem_inv_smul_set_iff₀ h _ _).mp h',
+end
+
+lemma balanced_core_aux_maximal {s t : set E} (h : t ⊆ s) (ht : balanced 𝕜 t) :
+  t ⊆ balanced_core_aux 𝕜 s :=
+begin
+  intros x hx,
+  rw balanced_core_aux_mem_iff,
+  intros r hr,
+  rw mem_smul_set_iff_inv_smul_mem₀ (norm_pos_iff.mp (lt_of_lt_of_le zero_lt_one hr)),
+  refine h (balanced_mem ht hx _),
+  rw norm_inv,
+  exact inv_le_one hr,
+end
+
+lemma balanced_core_subset_balanced_core_aux {s : set E} :
+  balanced_core 𝕜 s ⊆ balanced_core_aux 𝕜 s :=
+balanced_core_aux_maximal (balanced_core_subset s) (balanced_core_balanced s)
+
+lemma balanced_core_eq_Inter {s : set E} (hs : (0 : E) ∈ s) :
+  balanced_core 𝕜 s = ⋂ (r : 𝕜) (hr : 1 ≤ ∥r∥), r • s :=
+begin
+  rw ←balanced_core_aux,
+  refine subset_antisymm balanced_core_subset_balanced_core_aux _,
+  refine balanced.subset_core_of_subset (balanced_core_aux_balanced _) (balanced_core_aux_subset s),
+  refine mem_of_subset_of_mem balanced_core_subset_balanced_core_aux (balanced_core_zero_mem hs),
+end
+
+end normed_field
+
+end balanced_hull