A lower bound for the mean width of Voronoi polyhedra of unit ball packings in E3

Abstract.

In this note we prove that the mean width of any Voronoi cell of a packing of unit balls in E ³ is at least ${{\pi }\over {6\sqrt {6}\cdot {\rm arcsin}({{1}\over {\sqrt 3}})-\sqrt {6}\pi }}= 2.3264\dots .$As a comparison we mention that according to the Strong Dodecahedral Conjecture the mean width of any Voronoi cell of a packing of unit balls in E ³ must be at least as large as 2.3736... the mean width of a regular dodecahedron of inradius 1.