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[Merged by Bors] - feat(analysis/convex): Carathéodory's convexity theorem #2960

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[Merged by Bors] - feat(analysis/convex): Carathéodory's convexity theorem #2960

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semorrison
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theorem caratheodory (s : set E) :
  convex_hull s = ⋃ (t : finset E) (w : ↑t ⊆ s) (b : t.card ≤ findim ℝ E + 1), convex_hull ↑t

and more explicitly

theorem eq_center_mass_card_dim_succ_of_mem_convex_hull (s : set E) (x : E) (h : x ∈ convex_hull s) :
  ∃ (t : finset E) (w : ↑t ⊆ s) (b : t.card ≤ findim ℝ E + 1)
    (f : E → ℝ), (∀ y ∈ t, 0 ≤ f y) ∧ t.sum f = 1 ∧ t.center_mass f id = x

semorrison and others added 30 commits June 4, 2020 11:48
@semorrison
feat(analysis/convex): preparatory statement for caratheodory
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Simplify the proof
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all the boring bits
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some more work
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naming
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fix naming
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various progress
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mostly done with linear algebra API now
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comment
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Fill in some easy sorrys
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unsorry convex_coefficients
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almost there
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some more
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Removing sorrys
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Merge branch 'master' into caratheodory2
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copyright header
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merge
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Remove more sorrys
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Merge branch 'caratheodory2' of github.com:leanprover-community/mathl…
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…ib into caratheodory2
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@jcommelin jcommelin added the blocked-by-other-PR This PR depends on another PR which is still in the queue. A bot manages this label via PR comment. label Jun 11, 2020
@jcommelin jcommelin changed the title feat(analysis/convex): Carathéodory's convexity theorem feat(analysis/convex): Carathéodory's convexity theorem [deps: 3030] Jun 11, 2020
@semorrison semorrison removed the blocked-by-other-PR This PR depends on another PR which is still in the queue. A bot manages this label via PR comment. label Jun 11, 2020
@semorrison semorrison changed the title feat(analysis/convex): Carathéodory's convexity theorem [deps: 3030] feat(analysis/convex): Carathéodory's convexity theorem Jun 11, 2020
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@jcommelin, did you want to split any further?

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@semorrison We can easily split off one more PR about finite dimensionality. I think it makes sense. I'll do it right now.

@jcommelin jcommelin mentioned this pull request Jun 13, 2020
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Done in #3056 .

@jcommelin jcommelin added the blocked-by-other-PR This PR depends on another PR which is still in the queue. A bot manages this label via PR comment. label Jun 13, 2020
@jcommelin jcommelin changed the title feat(analysis/convex): Carathéodory's convexity theorem feat(analysis/convex): Carathéodory's convexity theorem [deps: 3056] Jun 13, 2020
@bryangingechen bryangingechen removed the blocked-by-other-PR This PR depends on another PR which is still in the queue. A bot manages this label via PR comment. label Jun 13, 2020
@bryangingechen bryangingechen changed the title feat(analysis/convex): Carathéodory's convexity theorem [deps: 3056] feat(analysis/convex): Carathéodory's convexity theorem Jun 13, 2020
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LGTM

src/analysis/convex/caratheodory.lean Outdated Show resolved Hide resolved
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bors merge

@github-actions github-actions bot added ready-to-merge All that is left is for bors to build and merge this PR. (Remember you need to say `bors r+`.) and removed awaiting-review The author would like community review of the PR labels Jun 16, 2020
bors bot pushed a commit that referenced this pull request Jun 16, 2020
@semorrison @urkud @jcommelin
feat(analysis/convex): Carathéodory's convexity theorem (#2960)
a432a3a 
```
theorem caratheodory (s : set E) :
  convex_hull s = ⋃ (t : finset E) (w : ↑t ⊆ s) (b : t.card ≤ findim ℝ E + 1), convex_hull ↑t
```
and more explicitly
```
theorem eq_center_mass_card_dim_succ_of_mem_convex_hull (s : set E) (x : E) (h : x ∈ convex_hull s) :
  ∃ (t : finset E) (w : ↑t ⊆ s) (b : t.card ≤ findim ℝ E + 1)
    (f : E → ℝ), (∀ y ∈ t, 0 ≤ f y) ∧ t.sum f = 1 ∧ t.center_mass f id = x
```



Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
Co-authored-by: Johan Commelin <johan@commelin.net>
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bors bot commented Jun 16, 2020

Pull request successfully merged into master.

Build succeeded:

@bors bors bot changed the title feat(analysis/convex): Carathéodory's convexity theorem [Merged by Bors] - feat(analysis/convex): Carathéodory's convexity theorem Jun 16, 2020
@bors bors bot closed this Jun 16, 2020
@bors bors bot deleted the caratheodory2 branch June 16, 2020 07:23
cipher1024 pushed a commit to cipher1024/mathlib that referenced this pull request Mar 15, 2022
@semorrison @urkud
@jcommelin @cipher1024
feat(analysis/convex): Carathéodory's convexity theorem (leanprover-c…
e6ea7a2 
…ommunity#2960)

```
theorem caratheodory (s : set E) :
  convex_hull s = ⋃ (t : finset E) (w : ↑t ⊆ s) (b : t.card ≤ findim ℝ E + 1), convex_hull ↑t
```
and more explicitly
```
theorem eq_center_mass_card_dim_succ_of_mem_convex_hull (s : set E) (x : E) (h : x ∈ convex_hull s) :
  ∃ (t : finset E) (w : ↑t ⊆ s) (b : t.card ≤ findim ℝ E + 1)
    (f : E → ℝ), (∀ y ∈ t, 0 ≤ f y) ∧ t.sum f = 1 ∧ t.center_mass f id = x
```



Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
Co-authored-by: Johan Commelin <johan@commelin.net>
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