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Minimal zonotopes containing the crosspolytope

by Henk, M., Linke, E., Wills, J. M..

Series: 2009-41, Preprints

52C07 Lattices and convex bodies in $n$ dimensions
11H06 Lattices and convex bodies

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 1-matrix, and that in each dimension the set of minimal zonotopes contains a parallelepiped. Based on that link to 1-matrices, we characterize all zonotopes attaining the minimal volume in
dimension 3 and present related results in higher dimensions.

zonotope, successive minima, Hadamard matrices

This paper was published in:
to appear in linear algebra and its applications