A HELLY-TYPE THEOREM FOR UNIONS OF CONVEX-SETS

We prove that for any d, k greater than or equal to 1 there are number s q = q(d, k) and h = Iz(d, k) such that the following holds: Let IC b e a family of sub sets of the d-dimensional Euclidean space, such that the intersection of any subfamily of K consisting of at most q sets c an be expressed as a union of at most k convex sets. Then the Helly nu mber of K is at most h. We also obtain topological generalizations of some cases oi this result. The main result was independently obtained by Alon and Kalai, by a different method.