Otto-von-Guericke-Universität Magdeburg

09-41

Minimal zonotopes containing the crosspolytope

Preprint series: 09-41, Preprints

The paper is published: to appear in linear algebra and its applications

MSC:
52C07 Lattices and convex bodies in $n$ dimensions, See Also {11H06, 11H31, 11P21}
Abstract: Motivated by the problem to improve Minkowski\'s lower bound on the successive minima for the class of zonotopes we determine the minimal volume of a zonotope containing the standard crosspolytope. It turns out that this volume can be expressed via the maximal determinant of a $\pm 1$-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to $\pm 1$-matrices, we characterize all zonotopes attaining the minimal volume in dimension 3 and present related results in higher dimensions.