On Lines Crossing Pairwise Intersecting Convex Sets in Three Dimensions

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.