PIERCING CONVEX-SETS

A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection. It is 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 F of compact, convex sets in R(d) that has the (p , q) property there is a set of at most c points in R(d) that intersects each member of F. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.