Abstract
It is widely believed that in a close packing of identical spheres in R3, the smallest Voronoi polyhedron is the regular dodecahedron. If this could be proved, the best upper bound on the density of such a packing would be reduced to 0.7547 …. Using a notion of local density due to Rogers, this paper proves that the optimal face of a Voronoi polyhedron is a regular pentagon slightly smaller than the pentagonal faces of the dodecahedron. This establishes an upper bound of 0.77836 … on the density, a marginal improvement on the previous best upper bound, recently proved by Lindsey. It is also shown that the best three-sided and four-sided faces are regular triangles and quadrilaterals, respectively.
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