Theorems · Theorem · convex and discrete geometry

convexIndependent_iff_notMem_convexHull_sdiff

∀ {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 ↔ ∀ (i : ι) (s : Set ι), p i ∉ (convexHull 𝕜) (p '' (s \ {i}))

If a family is convex independent, a point in the family is not in the convex hull of the other points. See convexIndependent_set_iff_notMem_convexHull_sdiff for the Set version.

Defined in

Mathlib.Analysis.Convex.Independent

Cited by

1 results in Mathlib

Foundations

Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPartialOrderAddCommGroupModule