A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection P-n of points on the plane contains two such that any circle containing them contains n/c elements of P-n, c a constant. We prove: Let Phi be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements S-i, S-j of Phi such that any set S' homothetic to S that contains them contains nle elements of Phi, c a constant (S' is homothetic to S if S' = lambda S + v, where lambda is a real number greater than 0 and v is a vector of R(2)). Our proof method is based on a new type of Voronoi diagram, called the ''closest covered set diagram'' based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Phi of n disjoint convex sets in R(3) such that for any nonempty subset Phi(H) of Phi there is a sphere S-H containing all the elements of Phi(H), and no other clement of Phi.