Theorems · Theorem · convex and discrete geometry

mem_convexHull_iff_exists_fintype

∀ {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} {x : E},
  x ∈ (convexHull R) s ↔ ∃ ι x_1 w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i, w i = 1 ∧ (∀ (i : ι), z i ∈ s) ∧ ∑ i, w i • z i = x

The convex hull of s is equal to the set of centers of masses of finite families of points in s. For universe reasons, you shouldn't use this lemma to prove that a given center of mass belongs to the convex hull. Use mem_convexHull_of_exists_fintype of the convex hull instead.

Defined in

Mathlib.Analysis.Convex.Combination

Cited by

2 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldAddCommGroupModuleLinearOrderIsStrictOrderedRing