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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Received April 14, 1995, and in revised form August 1, 1995.
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Matoušek, J. A Helly-Type Theorem for Unions of Convex Sets . Discrete Comput Geom 18, 1–12 (1997). https://doi.org/10.1007/PL00009305
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DOI: https://doi.org/10.1007/PL00009305