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The diameter of lattice zonotopes
Journal article

The diameter of lattice zonotopes

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 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.

Authors

Deza A; Pournin L; Sukegawa N

Journal

Proceedings of the American Mathematical Society, Vol. 148, No. 8, pp. 3507–3516

Publisher

American Mathematical Society (AMS)

Publication Date

March 30, 2020

DOI

10.1090/proc/14977

ISSN

0002-9939

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