A Crystalline Approximation Theorem for Hypersurfaces

The doctoral dissertation

of John M. Sullivan

Princeton University, 1990

My dissertation, A Crystalline Approximation Theorem for Hypersurfaces, shows that any hypersurface can be approximated arbitrarily well by polygons chosen from the finite set of facets of an appropriate cell complex, with restricted orientation and positions. Thus we can approximate the problem of finding the least-area surface on a given boundary by a finite network-flow problem in linear programming. This gives an effective algorithm for finding such surfaces, without knowing their topology in advance.

A slight extension of the main result of my thesis allows the minimization of energies which include bulk terms like volume or gravity as well as surface energy. This was published as Computing hypersurfaces which minimize surface energy plus bulk energy appeared in Motion by Mean Curvature and Related Topics (proceedings Trento 1992), de Gruyter, 1994, pp 186-197.

One lemma in my thesis concerned explicit bounds for the Besicovitch covering theorem. An expanded version was published as Sphere Packings Give an Explicit Bound for the Besicovitch Covering Theorem, J. Geometric Analysis 4:2, 1994, pp 219-231. This paper examines a standard proof of the Besicovitch Covering Theorem from the point of view of finding the optimal constant, which turns out to also be the answer to a sphere-packing problem: how many unit spheres fit into a ball of radius five? In high dimensions, I review the best asymptotic bounds known. In two dimensions, I show the answer is 19, while in three dimensions, I give the best upper and lower bounds known.