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
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The sphere packing problem asks for the densest packing of unit balls in E d . This problem has its roots in geometry, number theory and information theory and it is part of Hilbert’s 18th problem. One of the most attractive results on the sphere packing problem was proved by Rogers in 1958. It can be phrased as follows. Take a regular d-dimensional simplex of edge length 2 in E d and then draw a d-dimensional unit ball around each vertex of the simplex. Let σ d denote the ratio of the volume of the portion of the simplex covered by balls to the volume of the simplex. Then the volume of any Voronoi cell in a packing of unit balls in E d is at least ω d /σ d , where ω d denotes the volume of a d-dimensional unit ball. This has the immediate corollary that the density of any unit ball packing in E d is at most σ d . In 1978 Kabatjanskii and Levenštein improved this bound for large d. In fact, Rogers’ bound is the presently known best bound for 4 ≤ d ≤ 42, and above that the Kabatjanskii—Levenštein 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 ≥ 8 and improve the Kabatjanskii—Levenštein upper bound in small dimensions. Namely, we show that the volume of any Voronoi cell in a packing of unit balls in E d , d ≥ 8, is at least \(\omega _d /\hat \sigma _d\) and so the density of any unit ball packing in E d , d ≥ 8, is at most \(\hat \sigma _d\), where \(\hat \sigma _d\) is a geometrically well-defined quantity satisfying the inequality \(\hat \sigma _d < \sigma _d\) for all d≥ 8. We prove this by showing that the surface area of any Voronoi cell in a packing of unit balls in E d , d ≥ 8, is at least \((d \cdot \omega _d )/\hat \sigma _d\).