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 kk, the largest possible diameter δz(d,k)\delta _z(d,k) of a lattice zonotope contained in the hypercube [0,k]d[0,k]^d is uniquely achieved by a primitive zonotope. We obtain, as a consequence, that δz(d,k)\delta _z(d,k) grows like kd/(d+1)k^{d/(d+1)} up to an explicit multiplicative constant, when dd is fixed and kk goes to infinity, providing a new lower bound on the largest possible diameter of a lattice polytope contained in [0,k]d[0,k]^d.