Packing density

Definition

Let $n \ge 1$ and let $\operatorname{vol}$ be the Lebesgue measure on $\R^n$. Let $B_R$ be the closed ball of radius $R$ centred at $0$. A packing in $\R^n$ is a countable family $\mathcal P = (K_i)_{i \in I}$ of compact sets with pairwise disjoint interiors. The upper density of $\mathcal P$ is $$\delta(\mathcal P) = \limsup_{R \to \infty} \frac{\sum_{i \in I} \operatorname{vol}(K_i \cap B_R)}{\operatorname{vol}(B_R)},$$ with the limit superior; when the limit exists it is the density of $\mathcal P$. For a compact set $K$ with nonempty interior, the packing density of $K$ is $\delta(K) = \sup \delta(\mathcal P)$, over all packings $\mathcal P$ of congruent copies of $K$.