Large Finite Lattice Packings

A finite lattice packing of a centrally symmetric convex body K in ^d is a family C+K for a finite subset C of a packing lattice of K. For >0 the density (C;K,) is defined by (C;K,) = card C·V(K)/V(conv C+K). Assume that Cⁿ is the optimal packing with given n=card C, n large. It was known that conv Cⁿ is a segment if is less than the sausage radius _s (>0), and the inradius r(conv Cⁿ) tends to infinity with n if is greater than the critical radius _c (_s). We prove that if >_c in ^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 ³ that if _s<<_c, then conv Cⁿ is eventually planar and its shape is not too far from being a disc. As an example, we show that _s=_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 _s<_c holds even for general (not only lattice) packings.