Bounding the piercing number

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.

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