Discrete & Computational Geometry

, Volume 13, Issue 3–4, pp 245–256 | Cite as

Bounding the piercing number

  • N. Alon
  • G. Kalai
Article

Abstract

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
.

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Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • N. Alon
    • 1
  • G. Kalai
    • 2
  1. 1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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