  Gersho's conjecture in 3D asserts the asymptotic periodicity and structure of
the optimal centroidal Voronoi tessellation. This relatively simple
crystallization problem remains to date open. We prove bounds on the geometric
complexity of optimal centroidal Voronoi tessellations which, combined with an
approach introduced by Gruber in 2D, reduce the resolution of the 3D Gersho's
conjecture to a finite (albeit large) computation of an explicit convex problem
in finitely many variables.
