Theorems · Theorem · convex and discrete geometry

ConvexIndependent.mem_convexHull_iff

∀ {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommGroup E]
  [inst_3 : Module 𝕜 E] {p : ι → E},
  ConvexIndependent 𝕜 p → ∀ (s : Set ι) (i : ι), p i ∈ (convexHull 𝕜) (p '' s) ↔ i ∈ s

If a family is convex independent, a point in the family is in the convex hull of some of the points given by a subset of the index type if and only if the point's index is in this subset.

Defined in

Mathlib.Analysis.Convex.Independent

Cited by

1 results in Mathlib

Foundations

Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommGroupModule