Theorems · Theorem · convex and discrete geometry

Convex.radon_partition

• 1000+ list: Radon's theorem

∀ {ι : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : AddCommGroup E] [inst_4 : Module 𝕜 E] {f : ι → E},
  ¬AffineIndependent 𝕜 f → ∃ I, ((convexHull 𝕜) (f '' I) ∩ (convexHull 𝕜) (f '' Iᶜ)).Nonempty

Radon's theorem on convex sets. Any family f of affine dependent vectors contains a set I with the property that convex hulls of I and Iᶜ intersect nontrivially. In particular, any d + 2 points in a d-dimensional space can be partitioned this way, since they are affinely dependent (see finrank_vectorSpan_le_iff_not_affineIndependent).

Defined in

Mathlib.Analysis.Convex.Radon

Cited by

1 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupModule