A `converse' to the Constraint Lemma

Abstract

The main result is a direct proof of the implication $(LVK{F}_{k,3})\Rightarrow (L{T}_{3k-1,3})$ below. Consider the following statements: ($LVK{F}_{1,3}$) From any 11 points in ${\mathbb{R}}^3$ one can choose 3 pairwise disjoint triples whose convex hulls have a common point. ($LVK{F}_{k,3}$) From any $6k+5$ points in ${\mathbb{R}}^{3k}$ one can choose 3 pairwise disjoint sets each containing $2k+1$ points and whose convex hulls have a common point. ($L{T}_{2,3}$) Any 7 points in ${\mathbb{R}}^2$ can be decomposed into 3 subsets whose convex hulls have a common point. ($L{T}_{d,3}$) Any $2d+3$ points in ${\mathbb{R}}^d$ can be decomposed into 3 subsets whose convex hulls have a common point. This statements are true, but the meaning of the article is the direct derivation of one statement from another.

References in corpus (2)

• Counterexamples to the topological Tverberg conjecture
• On van Kampen-Flores, Conway-Gordon-Sachs and Radon theorems