Covering a convex 3D polytope by a minimal number of congruent spheres

Abstract

The problem of covering a convex 3D polytope by the minimal number of congruent spheres is reduced to a sequence of problems of minimising sphere radius when fixing the number of the spheres. We form a mathematical model of the problem using the Voronoi polytopes. Characteristics of the model are investigated. Extrema are attained at the vertices of the Voronoi polytopes constructed for sphere centres. To search for local minima, a modification of the Zoutendijk feasible directions method in a combination with random search is developed. Some numerical results for a cube and a non-regular octahedron are obtained.