Proof of a conjecture of B\'ar\'any, Katchalski and Pach

TL;DR

This paper confirms a conjecture regarding a quantitative version of Helly's theorem, establishing a bound on the volume of intersections of convex sets in Euclidean space that improves previous results.

Contribution

The paper proves the conjectured bound on the volume of intersections, advancing the understanding of convex set intersections in high-dimensional spaces.

Findings

Confirmed the conjecture v(d) ≤ d^{cd} for the volume bound

Improved the theoretical bound from previous results

Enhanced understanding of convex set intersection volumes

Abstract

B\'ar\'any, Katchalski and Pach proved the following quantitative form of Helly's theorem. If the intersection of a family of convex sets in $\mathbb{R}^d$ is of volume one, then the intersection of some subfamily of at most $2d$ members is of volume at most some constant $v(d)$. They proved the bound $v(d)\leq d^{2d^2}$, and conjectured $v(d)\leq d^{cd}$. We confirm it.