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Based on Minkowski's work on critical lattices of 3-dimensional convex bodies we present an eecient 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.

Hadwiger showed by computing the intrinsic volumes of a regular sim-plex that a rectangular simplex is a counterexample to Wills's conjecture for the relation between the lattice point enumerator and the intrinsic volumes in dimensions not less than 441. Here we give formulae for the volumes of spherical polytopes related to the intrinsic volumes of the… (More)

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 Lovász on covering minima.

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Here we show corresponding inequalities for arbitrary convex bodies, where the successive minima are replaced by certain successive diameters and successive widths. We further give some applications of these results to successive… (More)

We study the following generalization of the inradius: For a convex body K in the d-dimensional Euclidean space and a linear k-plane L we define the inradius of K with respect to L by rL(K) = max{r(K; x + L) : x ∈ E d }, where r(K; x + L) denotes the ordinary inradius of K ∩ (x + L) with respect to the affine plane x + L. We show how to determine rL(P) for… (More)

It is a well known fact that for every polynomial time algorithm which gives an upper bound V (K) and a lower bound V (K) for the volume of a convex set K ⊂ E d , the ratio V (K)/V (K) is at least (cd/ log d) d. Here we describe an algorithm which gives for ǫ > 0 in polynomial time an upper and lower bound with the property V (K)/V (K) ≤ d!(1 + ǫ) d .

We show that the sausage conjecture of László Fejes Tóth on finite sphere pack-ings is true in dimension 42 and above.