Bounds on the Geometric Complexity of Optimal Centroidal Voronoi Tesselations in 3D

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 as the number of generators tends to infinity. Combined with an approach of Gruber in 2D, these bounds reduce the resolution of the 3D Gersho’s conjecture to a finite, albeit very large, computation of an explicit convex problem in finitely many variables.