Multiple Sphere Packings

Abstract

A family S_n + X of unit spheres is said to form a k-fold packing in Eⁿ if each point of the space belongs to the interiors of at most k spheres of the family. In particular, when X is a lattice, the corresponding family is called a k-fold lattice packing. Let m (S_n, k, l) be the maximal number of unit spheres forming a k-fold packing and contained in lI_n. Then analogously to the densities of classical sphere packings we define

$$ \begin{gathered} {\mathbf{ }}\delta _k \left( {S_n } \right) = \mathop {\lim \sup }\limits_{l \to + \infty } \frac{{m\left( {S_n ,k,l} \right)v\left( {S_n } \right)}} {{v\left( {lI_n } \right)}} \hfill \\ {\text{and}} \hfill \\ {\mathbf{ }}\delta _{\text{k}}^{\text{*}} \left( {S_n } \right) = \mathop {\sup }\limits_\Lambda \frac{{v\left( {S_n } \right)}} {{\det \left( \Lambda \right)}}, \hfill \\ \end{gathered} $$

, where the supremum is over all lattices Λ such that S_n + Λ is a k-fold lattice packing in Eⁿ.