Bounding the piercing number
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- Alon, N. & Kalai, G. Discrete Comput Geom (1995) 13: 245. doi:10.1007/BF02574042
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It is shown that for everyk and everyp≥q≥d+1 there is ac=c(k,p,q,d)<∞ such that the following holds. For every familyℋ 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 ofℋ. It is also shown that for everyp≥q≥d+1 there is aC=C(p,q,d)<∞ such that, for every family 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.
© Springer-Verlag New York Inc. 1995