, Volume 13, Issue 1, pp 245-256

Bounding the piercing number

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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

http://static-content.springer.com/image/art%3A10.1007%2FBF02574042/MediaObjects/454_2007_BF02574042_f1.jpg
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
http://static-content.springer.com/image/art%3A10.1007%2FBF02574042/MediaObjects/454_2007_BF02574042_f2.jpg
.