Piercing intersecting convex sets

Abstract

Assume two finite families $\mathcal{A}$ and $\mathcal{B}$ of convex sets in ${\mathbb{R}}^3$ have the property that $A\cap B\ne \varnothing$ for every $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Is there a constant ${\hat{I}}^3>0$ (independent of $\mathcal{A}$ and $\mathcal{B}$) such that there is a line intersecting ${\hat{I}}^3|\mathcal{A}|$ sets in $\mathcal{A}$ or ${\hat{I}}^3|\mathcal{B}|$ sets in $\mathcal{B}$? This is an intriguing Helly-type question from a paper by Martínez, 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.

Accepted to Linear Algebra and its Applications