Discrete & Computational Geometry

, Volume 13, Issue 3, pp 245-256

First online:

Bounding the piercing number

  • N. AlonAffiliated withDepartment of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University
  • , G. KalaiAffiliated withInstitute of Mathematics, The Hebrew University of Jerusalem

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It is shown that for everyk and everypqd+1 there is ac=c(k,p,q,d)<∞ such that the following holds. For every family 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 of. It is also shown that for everypqd+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