Discrete & Computational Geometry

, Volume 13, Issue 3, pp 245–256

Bounding the piercing number

  • N. Alon
  • G. Kalai
Article

DOI: 10.1007/BF02574042

Cite this article as:
Alon, N. & Kalai, G. Discrete Comput Geom (1995) 13: 245. doi:10.1007/BF02574042

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 inRd 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 inRd 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
https://static-content.springer.com/image/art%3A10.1007%2FBF02574042/MediaObjects/454_2007_BF02574042_Fig1_HTML.jpg
of compact, convex sets inRd 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
https://static-content.springer.com/image/art%3A10.1007%2FBF02574042/MediaObjects/454_2007_BF02574042_Fig2_HTML.jpg
.

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