  We establish sharp asymptotic estimates for the diameter of primitive
zonotopes when their dimension is fixed. We also prove that, for infinitely
many integers $k$, the largest possible diameter of a lattice zonotope
contained in the hypercube $[0,k]^d$ is uniquely achieved by a primitive
zonotope. As a consequence, we obtain that this largest diameter 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 $[0,k]^d$.
