# A `converse' to the Constraint Lemma

## TL;DR

This paper provides a direct proof of a geometric implication relating to convex hull intersections in Euclidean spaces, connecting specific combinatorial configurations of points across different dimensions.

## Contribution

It offers a novel direct proof of the implication between two geometric lemmas, enhancing understanding of convex hull intersection properties.

## Key findings

- Proved the implication $(LVKF_{k,3})\Rightarrow( LT_{3k-1,3})$ directly.
- Confirmed the validity of the statements $LVKF_{k,3}$ and $LT_{d,3}$.
- Established a new proof technique for convex hull intersection lemmas.

## Abstract

The main result is a direct proof of the implication $(LVKF_{k,3})\Rightarrow( LT_{3k-1,3})$ below. Consider the following statements: ($LVKF_{1,3}$) From any 11 points in $ \mathbb{R}^{3}$ one can choose 3 pairwise disjoint triples whose convex hulls have a common point. ($LVKF_{k,3}$) 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. ($LT_{2,3}$) Any 7 points in $\mathbb{R}^{2}$ can be decomposed into 3 subsets whose convex hulls have a common point. ($LT_{d,3}$) 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.