Piercing intersecting convex sets

TL;DR

This paper investigates a Helly-type problem about intersecting convex sets in three-dimensional space, proving a special case where all sets in each family lie in parallel planes and establishing the existence of a line intersecting a significant portion of one family.

Contribution

The paper confirms a special case of a Helly-type question for intersecting convex sets in D, showing that under certain parallel plane conditions, a line intersects many sets in one family.

Findings

Confirmed the Helly-type property when all sets in each family lie in parallel planes.

Proved the existence of a line intersecting a large fraction of one family of convex sets.

Extended understanding of piercing problems for convex sets in three-dimensional space.

Abstract

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $\gamma >0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $\gamma|\mathcal A|$ sets in $\mathcal A$ or $\gamma|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Mart\'{i}nez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.