Large Finite Lattice Packings

Abstract

A finite lattice packing of a centrally symmetric convex body K in \(\mathbb{R}\)d is a family C+K for a finite subset C of a packing lattice Λ of K. For ϱ>0 the density δ (C;K,ϱ) is defined by δ(C;K,ϱ) = card C·V(K)/V(conv C+ϱK). Assume that Cn is the optimal packing with given n=card C, n large. It was known that conv Cn is a segment if ϱ is less than the sausage radius ϱs (>0), and the inradius r(conv Cn) tends to infinity with n if ϱ is greater than the critical radius ϱc (≥ϱs). We prove that if ϱ>ϱc in \(\mathbb{R}\)d, then the shape of conv Cn is not too far from being a ball. In addition, if r(conv Cn) is bounded but the radius of the largest (d−2)-ball in Cn tends to infinity, then eventually Cn is contained in some k–plane and its shape is not too far from being a k-ball where either k=d−1 or k=d−2. This yields in \(\mathbb{R}\)3 that if ϱs<ϱ<ϱc, then conv Cn is eventually planar and its shape is not too far from being a disc. As an example, we show that ϱs=ϱc if K is a 3-ball, verifying the Strong Sausage Conjecture in this case. On the other hand, if K is the octahedron then ϱs<ϱc holds even for general (not only lattice) packings.