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 C ⁿ is the optimal packing with given n=card C, n large. It was known that conv C ⁿ is a segment if ϱ is less than the sausage radius ϱ _s (>0), and the inradius r(conv C ⁿ) tends to infinity with n if ϱ is greater than the critical radius ϱ _c (≥ϱ _s ). We prove that if ϱ>ϱ _c in $\mathbb{R}$ ^d, then the shape of conv C ⁿ is not too far from being a ball. In addition, if r(conv C ⁿ) is bounded but the radius of the largest (d−2)-ball in C ⁿ tends to infinity, then eventually C ⁿ 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 ϱ _s <ϱ<ϱ _c , then conv C ⁿ is eventually planar and its shape is not too far from being a disc. As an example, we show that ϱ _s =ϱ _c if K is a 3-ball, verifying the Strong Sausage Conjecture in this case. On the other hand, if K is the octahedron then ϱ _s <ϱ _c holds even for general (not only lattice) packings.