BOUNDING THE PIERCING NUMBER

It is shown that for every k and every p greater than or equal to q greater than or equal to d + 1 there is a c = c(k, p, q, d) < infinity 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 greater than or equal to q greater than or equal to d + 1 there is a C = C(p, q, d)< infinity such that, for every family G 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 G.