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 $p \geq q \geq d + 1$ there is a $c = c(p,q,d) < \infty$ such that for every family $\mathcal{F}$ 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 $\mathcal{F}$. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.