The diameter of lattice zonotopes

Deza, Antoine; Pournin, Lionel; Sukegawa, Noriyoshi

doi:10.1090/proc/14977

The diameter of lattice zonotopes
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by Antoine Deza, Lionel Pournin and Noriyoshi Sukegawa PDF

Proc. Amer. Math. Soc. 148 (2020), 3507-3516 Request permission

Abstract:

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 $[0,k]^d$.

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