A Quantitative Helly-type Theorem: Containment in a Homothet

TL;DR

This paper introduces a new quantitative Helly-type theorem involving homothetic distance, establishing bounds on the diameter of intersections of convex sets with improved estimates over previous results.

Contribution

The paper presents a novel variant of Helly-type theorems based on homothetic distance and provides improved bounds for the diameter of intersections of convex sets.

Findings

Established a diameter bound of $(2d)^3 ext{diam}(K)$ for intersections of convex bodies.

Improved previous diameter estimate from $c d^{11/2}$ to a tighter bound.

Confirmed the conjectured lower bound on the multiplicative factor involving $d^{1/2}$.

Abstract

We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the \emph{diameter}. If $K$ is the intersection of finitely many convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d)^3\mathrm{diam}(K)$. The best previously known estimate, due to Brazitikos, is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$ conjectured by B\'ar\'any, Katchalski and Pach cannot be improved.