Minimal Surface Convex Hulls of Spheres

We present and solve a new computational geometry optimization problem. Spheres with given radii should be arranged such that (a) they do not overlap and (b) the surface area of the boundary of the convex hull enclosing the spheres is minimized. An additional constraint could be to fit the spheres into a specified geometry, e.g., a rectangular solid. To tackle the problem, we derive closed non-convex NLP models for this sphere arrangement or sphere packing problem. For two spheres, we prove that the minimal area of the boundary of the convex hull is identical to the sum of the surface areas of the two spheres. For special configurations of spheres we provide theoretical insights and we compute analytically minimal-area configurations. Numerically, we have solved problems containing up to 200 spheres.