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Improving Rogers' upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d>7

Karoly Bezdek
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Abstract

The sphere packing problem asks for the densest packing of unit balls in d-dimensional Euclidean space. This problem has its roots in geometry, number theory and it is part of Hilbert's 18th problem. In 1958 C. A. Rogers proved a non-trivial upper bound for the density of unit ball packings in d-dimensional Euclidean space for all d>0. In 1978 Kabatjanskii and Levenstein improved this bound for large d. In fact, Rogers' bound is the presently known best bound for 43>d>3, and above that the Kabatjanskii-Levenstein bound takes over. In this paper we improve Rogers' upper bound for the density of unit ball packings in Euclidean d-space for all d>7. We do this by estimating from below the surface area of Voronoi cells in any packing of unit balls in Euclidean d-space for all d>7.

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Karoly Bezdek. 2001-10-18. Improving Rogers' upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d>7. https://arxiv.org/abs/math/0110205

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