Abstract  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 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 S_i, S_j 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 Φ_Hh of Φ there is a sphere S_H containing all the elements of Φ_H, and no other element of Φ.