Bounding the piercing number

It is shown that for every k and every p≥q≥d+1 there is a c=c(k,p,q,d)<∞ such that the following holds. For every family ℋ whose members are unions of at most k compact convex sets in R ^d in which any set of p members of the family contains a subset of cardinality q with a nonempty intersection there is a set of at most c points in R ^d that intersects each member of ℋ. It is also shown that for every p≥q≥d+1 there is a C=C(p,q,d)<∞ such that, for every family[Figure not available: see fulltext.] of compact, convex sets in R ^d so that among and p of them some q have a common hyperplane transversal, there is a set of at most C hyperplanes that together meet all the members of[Figure not available: see fulltext.].