About an Erd\H{o}s-Gr\"unbaum conjecture concerning piercing of non bounded convex sets

TL;DR

This paper investigates conditions under which a family of convex sets in Euclidean space has a bounded piercing number, focusing on local intersection properties and their implications for the entire family.

Contribution

It provides new results on how local intersection conditions involving compact sets influence the global piercing number in convex set families.

Findings

Established bounds on piercing numbers based on local intersection properties.

Extended the analysis to variations of the intersection problem.

Answered a conjecture of Erd"H{o}s and Grünbaum regarding convex sets.

Abstract

In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erd\H{o}s and Gr\"unbaum. Namely, if in an infinite family of convex sets in $\mathbb{R}^d$ we know that out of every $p$ there are $q$ which are intersecting, we determine if having some compact sets implies a bound on the number of points needed to intersect the whole family. We also study variations of this problem.