A Helly-Type Theorem for Unions of Convex Sets

Discrete & Computational Geometry, Jul 1997

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 $${\cal K}$$ be a family of subsets of the d-dimensional Euclidean space, such that the intersection of any subfamily of $${\cal K}$$ consisting of at most q sets can be expressed as a union of at most k convex sets. Then the Helly number of $${\cal K}$$ 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.

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J. Matoušek. A Helly-Type Theorem for Unions of Convex Sets, Discrete & Computational Geometry, 1997, 1-12, DOI: 10.1007/PL00009305