Minimal zonotopes containing the crosspolytope

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.