Martin Henk

Densest lattice packings of 3-polytopes

Abstract Based on Minkowskis 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.

Randomized Simplex Algorithms on Klee-Minty Cubes

The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature.At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability.

Successive-minima-type inequalities

We show analogues of Minkowskis 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.

Notes on the Roots of Ehrhart Polynomials

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 n2, where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n2. 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 Ehrhart polyomials in the three-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.

A generalization of Jung's theorem

Jungs theorem establishes a relation between circumradius and diameter of a convex body. Half of the diameter can be interpreted as the maximum of circumradii of all 1-dimensional sections or 1-dimensional orthogonal projections of a convex body. This point of view leads to two series of j-dimensional circumradii, defined via sections or projections. In this paper we study some relations between these circumradii and by this we find a natural generalization of Jungs theorem.

Ehrhart Polynomials and Successive Minima

We investigate the Ehrhart polynomial for the class of 0-sym- metric convex lattice polytopes in Euclidean n-space R n . It turns out that the roots of the Ehrhart polynomial and Minkowskis successive minima are closely related by their geometric and arithmetic mean. We also show that the roots of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, we study the structure of the roots in the planar case. Here it turns out that their distribution reflects basic properties of lattice polygons.

Estimating sizes of a convex body by successive diameters and widths

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 radii, intrinsic volumes and the lattice point enumerator of a convex body.

Expected Frobenius numbers

Given a primitive positive integer vector a, the Frobenius number F(a) is the largest integer that cannot be represented as a non-negative integral combination of the coordinates of a. We show that for large instances the order of magnitude of the expected Frobenius number is (up to a constant depending only on the dimension) given by its lower bound.

Integer Knapsacks: Average Behavior of the Frobenius Numbers

The largest integer that cannot be represented as a nonnegative integral combination of given set of positive integers is called the Frobenius number of these integers. We show that the asymptotic growth of the Frobenius number on average is significantly slower than the growth of the maximum Frobenius number.

Approximating the volume of convex bodies

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