Lattice width directions and Minkowski's 3^d-theorem

We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864–12 January 1909), for which we provide two independent proofs.