Voronoi diagrams and containment of families of convex sets on the plane

Abstract

It is known that any family Pn of n points on the plane contains two elements such that any circle containing them contains *elements of Pn. We prove: Let @be a family of n disjoint compact convex sets on the plane, S be a strictly convex compact set. Then there are two elements Si, Sj of @ such that any set S’ homothetic to S that contains them contains ~ elements of 0, c a constant. 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 Y of n disjoint convex sets in ‘33 such that for any subset H of Y there is a sphere containing all of the elements of H, and no other element of Y-H is contained in it. Finally, we show using closed covered Voronoi diagrams that the ordered set B5 consisting of five bottom elements and ten top elements, one above each pair of bottoms, is not a circle order.