Density bounds for outer parallel domains of unit ball packings

We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.Comment: 22 pages, 1 figur

Periodicity and Circle Packing in the Hyperbolic Plane

We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius $r$-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius $r$-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius.Comment: 25 page

On a strong version of the Kepler conjecture

We raise and investigate the following problem that one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least 13.8564... .Comment: 9 page

On contact numbers of totally separable unit sphere packings

Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n>1 and d>1.Comment: 11 page

Sphere packings II

An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are divided into two classes. The first class of polyhedra, called quasi-regular tetrahedra, have density at most that of a regular tetrahedron. The polyhedra in the remaining class have density at most that of a regular octahedron (about 0.7209).Comment: 18 pages. Second of two older papers in the series on the proof of the Kepler conjecture. See math.MG/9811071. The original abstract is preserve