Improved Helly numbers of product sets

TL;DR

This paper improves bounds on the Helly number for product sets in Euclidean space, including exponential grids and integer sets defined by congruences, advancing understanding in convex geometry.

Contribution

It provides new bounds on the Helly number for product sets like exponential grids and integer congruence sets, extending classical Helly theory.

Findings

Improved bounds for Helly numbers of exponential grid product sets.

Extended Helly number bounds to integer sets with congruence relations.

Enhanced understanding of convex intersection properties in product sets.

Abstract

A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it exists, such that every $k$-intersecting family contains a point of $S$ in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form $A^d$ for various sets $A \subseteq \mathbb{R}$, including the ``exponential grid'' $A = \{\alpha^n : n \in \mathbb{N}\}$ and sets $A\subseteq \mathbb{Z}$ defined by congruence relations.