Minkowski Arrangements of Spheres

Abstract

Let μ be a non-negative number not greater than 1. Consider an arrangement ${\cal S}$ of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of ${\cal S}$ associate a concentric sphere of radius μ times the radius of S. We call this sphere the μ-kernel of S. The arrangement ${\cal S}$ is said to be a Minkowski arrangement of order μ if no sphere of ${\cal S}$ overlaps the μ-kernel of another sphere. The problem is to find the greatest possible density $d_n (\mu)$ of n-dimensional Minkowski sphere arrangements of order μ. In this paper we give upper bounds on $d_n (\mu)$ for $\mu \le {1 \over n}$ .