Application of an idea of Voronoĭ to lattice packing, supplement

Abstract

Voronoĭ’s criterion on lattice packings of balls in \(\mathbb E^d\) of local maximum density was extended in an earlier article to lattice packings of convex bodies in \(\mathbb E^d\) with local ultra maximum density, a stronger version of maximum density. No example of such a packing was given. However, it was conjectured that for generic convex bodies, the lattice packings of local maximum density are of local ultra maximum density provided that the dimension \(d\) is sufficiently high. For generic convex bodies, the kissing number of lattice packings of global maximum density is between \(d(d+1)\) and \(2d^2\). We conjecture that it equals \(2d^2\). In this note, first steps toward these conjectures are made. The existence of convex bodies with lattice packings of local ultra maximum density is shown. Contrasting this, families of convex bodies are specified which do not admit lattice packings of local ultra maximum density. If for a dense family of convex bodies every lattice packing of global maximum density has ultra maximum density, then this holds for a dense open family of convex bodies as well. A related result treats kissing numbers.