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

TL;DR

This paper establishes bounds on the geometric complexity of optimal 3D centroidal Voronoi tessellations, reducing Gersho's conjecture to a finite convex optimization problem, advancing understanding in geometric quantization.

Contribution

It provides the first bounds on the complexity of optimal 3D centroidal Voronoi tessellations, linking the problem to a finite convex optimization approach.

Findings

Bounds on the geometric complexity of optimal tessellations

Reduction of Gersho's conjecture to a finite convex problem

Potential for computational verification of the conjecture

Abstract

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.