Abstract
The geometric variant of a criterion of Voronoĭ says, a lattice packing of balls in \(\mathbb{E }^d\) has (locally) maximum density if and only if it is eutactic and perfect. This article deals with refinements of Voronoĭ’s result and extensions to lattice packings of smooth convex bodies. Versions of eutaxy and perfection are used to characterize lattices with semi-stationary, stationary, maximum and ultra-maximum lattice packing density, where ultra-maximality is a sharper version of maximality. Surprisingly, for balls, the lattice packings with maximum density have ultra-maximum density. To make the picture more complete, for \(d=2,3\), we specify the lattices that provide lattice packings of balls with maximum properties. These lattices are related to Bravais types. Finally, similar results of a duality type are given.
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