  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.
