File: Mathlib/Analysis/Convex/Hull.lean

full_name: convexHull_isClosed
code: /-- The convex hull of a set `s` is the minimal convex set that includes `s`. -/
@[simps! isClosed]
def convexHull : ClosureOperator (Set E) 
goal: (convexHull 𝕜).IsClosed s = Convex 𝕜 s

full_name: convexHull
code: /-- The convex hull of a set `s` is the minimal convex set that includes `s`. -/
@[simps! isClosed]
def convexHull : ClosureOperator (Set E) 
goal: ClosureOperator (Set E)

full_name: subset_convexHull
code: theorem subset_convexHull : s ⊆ convexHull 𝕜 s 
goal: s ⊆ (convexHull 𝕜) s

full_name: convex_convexHull
code: theorem convex_convexHull : Convex 𝕜 (convexHull 𝕜 s) 
goal: Convex 𝕜 ((convexHull 𝕜) s)

full_name: convexHull_eq_iInter
code: theorem convexHull_eq_iInter : convexHull 𝕜 s = ⋂ (t : Set E) (_ : s ⊆ t) (_ : Convex 𝕜 t), t 
goal: (convexHull 𝕜) s = ⋂ t, ⋂ (_ : s ⊆ t), ⋂ (_ : Convex 𝕜 t), t

full_name: mem_convexHull_iff
code: theorem mem_convexHull_iff : x ∈ convexHull 𝕜 s ↔ ∀ t, s ⊆ t → Convex 𝕜 t → x ∈ t 
goal: x ∈ (convexHull 𝕜) s ↔ ∀ (t : Set E), s ⊆ t → Convex 𝕜 t → x ∈ t

full_name: convexHull_min
code: theorem convexHull_min : s ⊆ t → Convex 𝕜 t → convexHull 𝕜 s ⊆ t 
goal: s ⊆ t → Convex 𝕜 t → (convexHull 𝕜) s ⊆ t

full_name: Convex.convexHull_subset_iff
code: theorem Convex.convexHull_subset_iff (ht : Convex 𝕜 t) : convexHull 𝕜 s ⊆ t ↔ s ⊆ t 
goal: (convexHull 𝕜) s ⊆ t ↔ s ⊆ t

full_name: convexHull_mono
code: @[mono, gcongr]
theorem convexHull_mono (hst : s ⊆ t) : convexHull 𝕜 s ⊆ convexHull 𝕜 t 
goal: (convexHull 𝕜) s ⊆ (convexHull 𝕜) t

full_name: convexHull_eq_self
code: lemma convexHull_eq_self : convexHull 𝕜 s = s ↔ Convex 𝕜 s 
goal: (convexHull 𝕜) s = s ↔ Convex 𝕜 s

full_name: convexHull_univ
code: @[simp]
theorem convexHull_univ : convexHull 𝕜 (univ : Set E) = univ 
goal: (convexHull 𝕜) Set.univ = Set.univ

full_name: convexHull_empty
code: @[simp]
theorem convexHull_empty : convexHull 𝕜 (∅ : Set E) = ∅ 
goal: (convexHull 𝕜) ∅ = ∅

full_name: convexHull_empty_iff
code: @[simp]
theorem convexHull_empty_iff : convexHull 𝕜 s = ∅ ↔ s = ∅ 
goal: (convexHull 𝕜) s = ∅ ↔ s = ∅

full_name: convexHull_nonempty_iff
code: @[simp]
theorem convexHull_nonempty_iff : (convexHull 𝕜 s).Nonempty ↔ s.Nonempty 
goal: ((convexHull 𝕜) s).Nonempty ↔ s.Nonempty

full_name: segment_subset_convexHull
code: theorem segment_subset_convexHull (hx : x ∈ s) (hy : y ∈ s) : segment 𝕜 x y ⊆ convexHull 𝕜 s 
goal: segment 𝕜 x y ⊆ (convexHull 𝕜) s

full_name: convexHull_singleton
code: @[simp]
theorem convexHull_singleton (x : E) : convexHull 𝕜 ({x} : Set E) = {x} 
goal: (convexHull 𝕜) {x} = {x}

full_name: convexHull_zero
code: @[simp]
theorem convexHull_zero : convexHull 𝕜 (0 : Set E) = 0 
goal: (convexHull 𝕜) 0 = 0

full_name: convexHull_pair
code: @[simp]
theorem convexHull_pair (x y : E) : convexHull 𝕜 {x, y} = segment 𝕜 x y 
goal: (convexHull 𝕜) {x, y} = segment 𝕜 x y

full_name: convexHull_convexHull_union_left
code: theorem convexHull_convexHull_union_left (s t : Set E) :
    convexHull 𝕜 (convexHull 𝕜 s ∪ t) = convexHull 𝕜 (s ∪ t) 
goal: (convexHull 𝕜) ((convexHull 𝕜) s ∪ t) = (convexHull 𝕜) (s ∪ t)

full_name: convexHull_convexHull_union_right
code: theorem convexHull_convexHull_union_right (s t : Set E) :
    convexHull 𝕜 (s ∪ convexHull 𝕜 t) = convexHull 𝕜 (s ∪ t) 
goal: (convexHull 𝕜) (s ∪ (convexHull 𝕜) t) = (convexHull 𝕜) (s ∪ t)

full_name: Convex.convex_remove_iff_not_mem_convexHull_remove
code: theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex 𝕜 s) (x : E) :
    Convex 𝕜 (s \ {x}) ↔ x ∉ convexHull 𝕜 (s \ {x}) 
goal: Convex 𝕜 (s \ {x}) ↔ x ∉ (convexHull 𝕜) (s \ {x})

full_name: IsLinearMap.image_convexHull
code: theorem IsLinearMap.image_convexHull {f : E → F} (hf : IsLinearMap 𝕜 f) (s : Set E) :
    f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) 
goal: f '' (convexHull 𝕜) s = (convexHull 𝕜) (f '' s)

full_name: LinearMap.image_convexHull
code: theorem LinearMap.image_convexHull (f : E →ₗ[𝕜] F) (s : Set E) :
    f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) 
goal: ⇑f '' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f '' s)

full_name: convexHull_smul
code: theorem convexHull_smul (a : 𝕜) (s : Set E) : convexHull 𝕜 (a • s) = a • convexHull 𝕜 s 
goal: (convexHull 𝕜) (a • s) = a • (convexHull 𝕜) s

full_name: AffineMap.image_convexHull
code: theorem AffineMap.image_convexHull (f : E →ᵃ[𝕜] F) (s : Set E) :
    f '' convexHull 𝕜 s = convexHull 𝕜 (f '' s) 
goal: ⇑f '' (convexHull 𝕜) s = (convexHull 𝕜) (⇑f '' s)

full_name: convexHull_subset_affineSpan
code: theorem convexHull_subset_affineSpan (s : Set E) : convexHull 𝕜 s ⊆ (affineSpan 𝕜 s : Set E) 
goal: (convexHull 𝕜) s ⊆ ↑(affineSpan 𝕜 s)

full_name: affineSpan_convexHull
code: @[simp]
theorem affineSpan_convexHull (s : Set E) : affineSpan 𝕜 (convexHull 𝕜 s) = affineSpan 𝕜 s 
goal: affineSpan 𝕜 ((convexHull 𝕜) s) = affineSpan 𝕜 s

full_name: convexHull_neg
code: theorem convexHull_neg (s : Set E) : convexHull 𝕜 (-s) = -convexHull 𝕜 s 
goal: (convexHull 𝕜) (-s) = -(convexHull 𝕜) s

full_name: convexHull_vadd
code: lemma convexHull_vadd (x : E) (s : Set E) : convexHull 𝕜 (x +ᵥ s) = x +ᵥ convexHull 𝕜 s 
goal: (convexHull 𝕜) (x +ᵥ s) = x +ᵥ (convexHull 𝕜) s