Bounding the piercing number |
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Authors: | N. Alon G. Kalai |
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Affiliation: | (1) Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel;(2) Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel |
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Abstract: | It is shown that for everyk and everyp≥q≥d+1 there is ac=c(k,p,q,d)<∞ such that the following holds. For every familyℋ whose members are unions of at mostk compact convex sets inR d in which any set ofp members of the family contains a subset of cardinalityq with a nonempty intersection there is a set of at mostc points inR d that intersects each member ofℋ. It is also shown that for everyp≥q≥d+1 there is aC=C(p,q,d)<∞ such that, for every family of compact, convex sets inR d so that among andp of them someq have a common hyperplane transversal, there is a set of at mostC hyperplanes that together meet all the members of . This research was supported in part by a United States-Israel BSF Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences. |
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