On a Helly-type question for central symmetry

TL;DR

This paper explores a Helly-type problem in convex geometry, examining whether large subsets with centrally symmetric convex positions imply the entire set is centrally symmetric, providing partial results for this conjecture.

Contribution

It investigates a Helly-type question for central symmetry, offering new partial results and insights into when a set must be centrally symmetric based on its subsets.

Findings

The property holds for small subset sizes (k ≤ 5).

Partial results suggest the property may hold for larger k.

The paper advances understanding of symmetry conditions in convex sets.

Abstract

We study a certain Helly-type question by Konrad Swanepoel. Assume that $X$ is a set of points such that every $k$-subset of $X$ is in centrally symmetric convex position, is it true that $X$ must also be in centrally symmetric convex position? It is easy to see that this is false if $k\le 5$, but it may be true for sufficiently large $k$. We investigate this question and give some partial results.