On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions

Abstract

The 1913 Helly's theorem states that any family $\mathcal{K}$ of $n\ge d+1$ convex sets in ${\mathbb{R}}^d$ can be pierced by a single point if and only if any $d+1$ of $\mathcal{K}$'s elements can. In 2002 Alon, Kalai, Matoušek and Meshulam ruled out the possibility of similar criteria for the existence of lines crossing multiple convex sets in dimension $d\ge 3$ -- for any $k\ge 3$, they described arbitrary large families $\mathcal{K}$ of convex sets in ${\mathbb{R}}^3$ so that any $k$ elements of $\mathcal{K}$ can be crossed by a line yet no $k+4$ of them can. Let $\mathcal{K}$ be a family of $n$ pairwise intersecting convex sets in ${\mathbb{R}}^3$. We show that there exists a line crossing $\Theta (n)$ elements of $\mathcal{K}$. This resolves the most extensively studied variant of a problem by Martínez, Roldán-Pensado and Rubin (Discrete Comput. Geom. 2020) which was highlighted by Bárány and Kalai (Bull. Amer. Math. Soc. 2021). Our result adds to the very few sufficient (and non-trivial) conditions that have been known for the existence of line transversals to large families of convex sets. Our argument is based on a Ramsey-type result of independent interest for families of pairwise intersecting convex sets in ${\mathbb{R}}^2$, and the structure of line arrangements in ${\mathbb{R}}^3$.

arXiv commentA preliminary version appeared in the Proceeding of SODA 2026