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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.

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 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.

We consider the homogenized linear feasibility problem, to find an x on the unit sphere, satisfying n linear inequalities a T i x ≥ 0. To solve this problem we consider the centers of the insphere of spherical sim-plices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algorithm for the linear feasibility… (More)

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

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