No abstract

We show analogues of Minkowski's theorem on successive minima, where the volume is replaced by the lattice point enumerator. We further give analogous results to some recent theorems by Kannan and Lov/tsz on covering minima.

Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an efficient algorithm for computing the density of a densest lattice packing of an arbitrary 3-polytope. As an application we calculate densest lattice packings of all regular and Archimedean polytopes. * The second named author acknowledges the hospitality of the International Erwin Schrödinger Institute for Mathematical Physics in Vienna, where a main part of his contribution to this work has been completed.