Holes in Sphere Packings

Abstract

In order to study the efficiency of a sphere packing, it is both important and interesting to investigate its holes, especially spherical holes. Let S _n + X be a sphere packing in Eⁿ, and let r(S _n , X) be the supremum of the radii of all spheres disjoint from any sphere of the packing. The number 1/r(S _n , X) is called the closeness of the packing S _n + X. We define

$$ r\left( {S_n } \right) = \mathop {\inf }\limits_X \left\{ {r\left( {S_n ,X} \right)} \right\}, $$

, where the infimum is taken over all sets X such that S _n + X is a packing. By routine argument it follows that there exists a discrete set X such that r(S _n ) = r(S _n , X).