Theorems · Theorem · convex and discrete geometry

Caratheodory.mem_convexHull_erase

∀ {𝕜 : Type u_1} {E : Type u} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : AddCommGroup E] [inst_4 : Module 𝕜 E] [inst_5 : DecidableEq E] {t : Finset E},
  ¬AffineIndependent 𝕜 Subtype.val → ∀ {x : E}, x ∈ (convexHull 𝕜) ↑t → ∃ y, x ∈ (convexHull 𝕜) ↑(t.erase ↑y)

If x is in the convex hull of some finset t whose elements are not affine-independent, then it is in the convex hull of a strict subset of t.

Defined in

Mathlib.Analysis.Convex.Caratheodory

Cited by

1 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupModuleDecidableEq