The optimal centroidal Voronoi tessellations and the gershoʹs conjecture in the three-dimensional space

Optimal centroidal Voronoi tessellations have important applications in many different areas such as vector quantization, data and image processing, clustering analysis, and resource management. In the three-dimensional Euclidean space, they are also useful to the mesh generation and optimization. In this paper, we conduct extensive numerical simulations to investigate the asymptotic structures of optimal centroidal Voronoi tessellations for a given domain. Such a problem is intimately related to the famous Gershoʹs conjecture, for which a full proof is still not available. We provide abundant evidence to substantiate the claim of the conjecture: the body-centered-cubic lattice (or Par6) based centroidal Voronoi tessellation has the lowest cost (or energy) per unit volume and is the most likely congruent cell predicted by the three-dimensional Gersho conjecture. More importantly, we probe the various properties of this optimal configuration including its dual triangulations which bear significant consequences in applications to three-dimensional high quality meshing.