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 8 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 8 such that any set S′ homothetic to S that contains them contains n/c elements of 8, c a constant (S′ is homothetic to S if S′ = λS+v, where λ is a real number greater than 0 and v is a vector of <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 8 of n disjoint convex sets in <3 such that for any nonempty subset 8H of 8 there is a sphere SH containing all the elements of 8H , and no other element of 8.