File: Mathlib/Analysis/Convex/Radon.lean

full_name: Convex.radon_partition
code: /-- **Radon's theorem on convex sets**.

Any family `f` of affine dependent vectors contains a set `I` with the property that convex hulls of
`I` and `Iᶜ` intersect nontrivially. -/
theorem radon_partition {f : ι → E} (h : ¬ AffineIndependent 𝕜 f) :
    ∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty 
goal: ∃ I, ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)).Nonempty

full_name: _private.Mathlib.Analysis.Convex.Radon.0.Convex.helly_theorem_corner
code: /-- Corner case for `helly_theorem'`. -/
private lemma helly_theorem_corner {F : ι → Set E} {s : Finset ι}
    (h_card_small : s.card ≤ finrank 𝕜 E + 1)
    (h_inter : ∀ I ⊆ s, I.card ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
    (⋂ i ∈ s, F i).Nonempty 
goal: (⋂ i ∈ s, F i).Nonempty

full_name: Convex.helly_theorem'
code: /-- **Helly's theorem** for finite families of convex sets.

If `F` is a finite family of convex sets in a vector space of finite dimension `d`, and any
`k ≤ d + 1` sets of `F` intersect nontrivially, then all sets of `F` intersect nontrivially. -/
theorem helly_theorem' {F : ι → Set E} {s : Finset ι}
    (h_convex : ∀ i ∈ s, Convex 𝕜 (F i))
    (h_inter : ∀ I ⊆ s, I.card ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
    (⋂ i ∈ s, F i).Nonempty 
goal: (⋂ i ∈ s, F i).Nonempty

full_name: Convex.helly_theorem
code: /-- **Helly's theorem** for finite families of convex sets in its classical form.

If `F` is a family of `n` convex sets in a vector space of finite dimension `d`, with `n ≥ d + 1`,
and any `d + 1` sets of `F` intersect nontrivially, then all sets of `F` intersect nontrivially. -/
theorem helly_theorem {F : ι → Set E} {s : Finset ι}
    (h_card : finrank 𝕜 E + 1 ≤ s.card)
    (h_convex : ∀ i ∈ s, Convex 𝕜 (F i))
    (h_inter : ∀ I ⊆ s, I.card = finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
    (⋂ i ∈ s, F i).Nonempty 
goal: (⋂ i ∈ s, F i).Nonempty

full_name: Convex.helly_theorem_set'
code: /-- **Helly's theorem** for finite sets of convex sets.

If `F` is a finite set of convex sets in a vector space of finite dimension `d`, and any `k ≤ d + 1`
sets from `F` intersect nontrivially, then all sets from `F` intersect nontrivially. -/
theorem helly_theorem_set' {F : Finset (Set E)}
    (h_convex : ∀ X ∈ F, Convex 𝕜 X)
    (h_inter : ∀ G : Finset (Set E), G ⊆ F → G.card ≤ finrank 𝕜 E + 1 → (⋂₀ G : Set E).Nonempty) :
    (⋂₀ (F : Set (Set E))).Nonempty 
goal: (⋂₀ ↑F).Nonempty

full_name: Convex.helly_theorem_set
code: /-- **Helly's theorem** for finite sets of convex sets in its classical form.

If `F` is a finite set of convex sets in a vector space of finite dimension `d`, with `n ≥ d + 1`,
and any `d + 1` sets from `F` intersect nontrivially,
then all sets from `F` intersect nontrivially. -/
theorem helly_theorem_set {F : Finset (Set E)}
    (h_card : finrank 𝕜 E + 1 ≤ F.card)
    (h_convex : ∀ X ∈ F, Convex 𝕜 X)
    (h_inter : ∀ G : Finset (Set E), G ⊆ F → G.card = finrank 𝕜 E + 1 → (⋂₀ G : Set E).Nonempty) :
    (⋂₀ (F : Set (Set E))).Nonempty 
goal: (⋂₀ ↑F).Nonempty

full_name: Convex.helly_theorem_compact'
code: /-- **Helly's theorem** for families of compact convex sets.

If `F` is a family of compact convex sets in a vector space of finite dimension `d`, and any
`k ≤ d + 1` sets of `F` intersect nontrivially, then all sets of `F` intersect nontrivially. -/
theorem helly_theorem_compact' [TopologicalSpace E] [T2Space E] {F : ι → Set E}
    (h_convex : ∀ i, Convex 𝕜 (F i)) (h_compact : ∀ i, IsCompact (F i))
    (h_inter : ∀ I : Finset ι, I.card ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
    (⋂ i, F i).Nonempty 
goal: (⋂ i, F i).Nonempty

full_name: Convex.helly_theorem_compact
code: /-- **Helly's theorem** for families of compact convex sets in its classical form.

If `F` is a (possibly infinite) family of more than `d + 1` compact convex sets in a vector space of
finite dimension `d`, and any `d + 1` sets of `F` intersect nontrivially,
then all sets of `F` intersect nontrivially. -/
theorem helly_theorem_compact [TopologicalSpace E] [T2Space E] {F : ι → Set E}
    (h_card : finrank 𝕜 E + 1 ≤ PartENat.card ι)
    (h_convex : ∀ i, Convex 𝕜 (F i)) (h_compact : ∀ i, IsCompact (F i))
    (h_inter : ∀ I : Finset ι, I.card = finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) :
    (⋂ i, F i).Nonempty 
goal: (⋂ i, F i).Nonempty

full_name: Convex.helly_theorem_set_compact'
code: /-- **Helly's theorem** for sets of compact convex sets.

If `F` is a set of compact convex sets in a vector space of finite dimension `d`, and any
`k ≤ d + 1` sets from `F` intersect nontrivially, then all sets from `F` intersect nontrivially. -/
theorem helly_theorem_set_compact' [TopologicalSpace E] [T2Space E] {F : Set (Set E)}
    (h_convex : ∀ X ∈ F, Convex 𝕜 X) (h_compact : ∀ X ∈ F, IsCompact X)
    (h_inter : ∀ G : Finset (Set E), (G : Set (Set E)) ⊆ F → G.card ≤ finrank 𝕜 E + 1 →
    (⋂₀ G : Set E).Nonempty) :
    (⋂₀ (F : Set (Set E))).Nonempty 
goal: (⋂₀ F).Nonempty

full_name: Convex.helly_theorem_set_compact
code: /-- **Helly's theorem** for sets of compact convex sets in its classical version.

If `F` is a (possibly infinite) set of more than `d + 1` compact convex sets in a vector space of
finite dimension `d`, and any `d + 1` sets from `F` intersect nontrivially,
then all sets from `F` intersect nontrivially. -/
theorem helly_theorem_set_compact [TopologicalSpace E] [T2Space E] {F : Set (Set E)}
    (h_card : finrank 𝕜 E + 1 ≤ F.encard)
    (h_convex : ∀ X ∈ F, Convex 𝕜 X) (h_compact : ∀ X ∈ F, IsCompact X)
    (h_inter : ∀ G : Finset (Set E), (G : Set (Set E)) ⊆ F → G.card = finrank 𝕜 E + 1 →
    (⋂₀ G : Set E).Nonempty) :
    (⋂₀ (F : Set (Set E))).Nonempty 
goal: (⋂₀ F).Nonempty