On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions

The 1913 Helly’s theorem states that any family K of n ≥ d + 1 convex sets in ℝ^d can be pierced by a single point if and only if any d+ 1 of 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 ≥ 3 – for any k ≥ 3, they described arbitrary large families K of convex sets in ℝ³ so that any k elements of K can be crossed by a line yet no k + 4 of them can. Let K be a family of n pairwise intersecting convex sets in ℝ³. We show that there exists a line crossing Θ(n) elements of K. This resolves the most extensively studied variant of a problem by Martinez, 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.