, Volume 13, Issue 1, pp 245-256

Bounding the piercing number

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 .

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.