Finite Lattice Packings and the Wulff‐Shape

Abstract

This paper treats finite lattice packings Cn + K of n copies of some centrally symmetric convex body K in Ed for large n. Assume that Cn is a subset of a lattice Λ, and ϱ is at least the covering radius; namely, Λ + ϱK covers the space. The parametric density δ(Cn, ϱ) is defined by δ(Cn, ϱ) = n · V(K)/V(convCn + ϱK). It is shown that, if δ(Cn, ϱ) is minimal for n large, then the shape of conv Cn is approximately given by Wulff's condition, well-known from crystallography. Thus maximizing parametric density is equivalent to optimizing a certain Gibbs–Curie energy. It is also proved that, in case of lattice packings of K (allowing any packing lattice), for large n the optimal shape with respect to the parametric density is approximately a Wulff-shape associated to some densest packing lattice of K.