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 .

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