Given n and a convex body K, we consider arrangements of n congruent copies, namely, packings inside convex containers and coverings of compact convex sets. Concerning density, clusters are naturally related to periodic arrangements (see Theorem 7.1.1). In addition, we verify that an asymptotic density exists as n tends to infinity (see Theorem 7.2.2), and we characterize the case when this asymptotic density is one (see Lemma 7.2.3). Example 7.2.4 shows that the asymptotic structure of the optimal arrangement depends very much on K.

Concerning the mean i-dimensional projection for i = 1, …, d − 1, we verify that the optimal convex hull of n nonoverlapping congruent copies of K is close to being a ball (see Theorem 7.2.2). In contrast, the compact convex set of maximal mean width covered by n congruent copies of K is close to being a segment (see Theorem 7.4.1).

Periodic and Finite Arrangements

Let K be a convex body in ℝd. If K is a periodic arrangement by congruent copies of K with respect to the lattice Λ (see Section A.13), and the number of equivalence classes is m, then the density of the arrangement is m · V(K)/det Λ. We define the packing density δ(K) to be supremum of the densities of periodic packings by congruent copies of K, and we let Δ(K) = V(K)/δ(K).