### A Crystalline Approximation Theorem for Hypersurfaces

#### The doctoral dissertation

#### 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.