A nonsmooth program for jamming hard spheres

Abstract

We study packings of $n$ hard spheres of equal radius in the $d$-dimensional unit cube. We present a nonsmooth function whose local extrema are the radii of jammed packings (where no subset of spheres can be moved keeping all others fixed) and show that for a fixed number of spheres there are only finitely many radii of such jammed configurations. We propose an algorithm for the maximization of this maximal radius function and present examples for 5 - 8 disks in the unit square and 4 - 6 spheres in the unit cube. The method allows straightforward generalization to packings of spheres in other compact containers.