Proofs on Unit Sphere Packings

Abstract

The proofs presented in this chapter can be grouped as follows. We prove lower and upper estimates for the contact numbers of packings of n unit balls in Euclidean 3-space. One can regard this problem as a combinatorial relative of the Kepler problem on the densest unit sphere packings. Next, we give lower estimates for the surface volume of Voronoi cells in packings of unit balls in Euclidean d-space for all d ≥ 2 and then we improve those estimates in dimensions d ≥ 8. All these results imply upper bounds for the usual density of unit ball packings. Returning to the 3-dimensional Euclidean space we give lower bounds for the average surface area (resp., average edge curvature) of the cells in an arbitrary normal tiling with each cell holding a unit ball. On the one hand, it leads to a new version of the Kepler problem on unit sphere packings on the other hand, it generates a new relative of Kelvin’s foam problem. Finally, we find sufficient conditions for sphere packings being uniformly stable, a property that holds for all densest lattice sphere packings up to dimension 8.KeywordsUnit BallVoronoi CellKepler ProblemRigidity MatrixSurface VolumeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.