A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn, c a constant. We prove: Let Φ 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 Si, Sj of Φ such that any set S′ homothetic to S that contains them contains n/c elements of Φ, c a constant (S is homothetic to S if 5’ = λS + v, where λ is a real number greater than 0 and v is a vector of ℜ2). 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 Φ of n disjoint convex sets in ℜ3)3 such that for any nonempty subset ΦHh of Φ there is a sphere SH containing all the elements of ΦH, and no other element of Φ.