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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.

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.

We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body.2010 Mathematics Subject Classification. 52A40, 52A38.

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Abstract. We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n 2 , where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun [8] where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n 2 .For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes.We further give a characterisation of the roots of the Ehrhart polyomials in the 3-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -along with its equality conditions. 2010 Mathematics Subject Classification. 52A40, 52B11.

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