The diameter of lattice zonotopes.

We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed and the number of their generators grows large. We also prove that, for infinitely many integers k, the largest possible diameter delta(z)(d, k) of a lattice zonotope contained in the hypercube [0, k](d) is uniquely achieved by a primitive zonotope. We obtain, as a consequence, that delta(z)(d, k) grows like k(d/(d+1)) up to an explicit multiplicative constant, when d is fixed and k goes to infinity, providing a new lower bound on the largest possible diameter of a lattice polytope contained in [0, k](d).