A helly-type theorem for unions of convex sets

Abstract

We prove that for any d, k ≥ 1 there are numbers q = q(d, k) and h = h(d, k) such that the following holds: Let K be a family of subsets of the d-dimensional Euclidean space, such that the intersection of any subfamily of K consisting of at most q sets can be expressed as a union of at most k convex sets. Then the Helly number of K is at most h. We also obtain topological generalizations of some cases of this result. The main result was independently obtained by Alon and Kalai, by a different method.