Karoly Bezdek - 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.