Theorems · Theorem · convex and discrete geometry

convexHull_eq

∀ {R : Type u_1} {E : Type u_3} [inst : Field R] [inst_1 : AddCommGroup E] [inst_2 : Module R E]
  [inst_3 : LinearOrder R] [IsStrictOrderedRing R] (s : Set E),
  (convexHull R) s = {x | ∃ ι t w z, (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x}

Convex hull of s is equal to the set of all centers of masses of Finsets t, z '' t ⊆ s. For universe reasons, you shouldn't use this lemma to prove that a given center of mass belongs to the convex hull. Use convexity of the convex hull instead.

Defined in

Mathlib.Analysis.Convex.Combination

Cited by

3 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldAddCommGroupModuleLinearOrderIsStrictOrderedRing