|KAROLY BEZDEK, Department of Geometry, Eotvos University, H-1088 Budapest, Hungary and Department of Mathematics, Cornell University Ithaca, New York 14853-7901, USA|
|On a stronger form of Rogers' lemma and the minimum surface area of Voronoi cells in unit ball packings|
Rogers' lemma is an essential tool for estimating the density of unit ball packings in Euclidean space. Also, it motivates several other techniques of the classical theory of packing. In this talk we prove a strengthening of Rogers' lemma and apply it to estimate the minimum surface area of Voronoi cells in unit ball packings. We prove a lower bound for the surface area of Voronoi cells of unit ball packings in d-dimensional Euclidean space. This bound is sharp for d=2 and implies Rogers' upper bound for the density of unit ball packings in d-space for all d>1. Finally, we strengthen these results for d=3.