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The sphere packing problem asks for the densest packing of unit balls ind-dimensional Euclidean space. This problem has its roots in geometry, numbertheory and it is part of Hilbert's 18th problem. In 1958 C. A. Rogers proved anon-trivial upper bound for the density of unit ball packings in d-dimensionalEuclidean space for all d>0. In 1978 Kabatjanskii and Levenstein improved thisbound for large d. In fact, Rogers' bound is the presently known best bound for43>d>3, and above that the Kabatjanskii-Levenstein bound takes over. In thispaper we improve Rogers' upper bound for the density of unit ball packings inEuclidean d-space for all d>7. We do this by estimating from below the surfacearea of Voronoi cells in any packing of unit balls in Euclidean d-space for alld>7.

discrete and computational geometry/discrete and computational geometry

Improving Rogers’ Upper Bound for the Density of Unit Ball Packings via Estimating the Surface Area of Voronoi Cells from Below in Euclidean \sl d -Space for All \sl d ≥ \bf 8