# The diameter of lattice zonotopes

## TL;DR

This paper provides sharp asymptotic estimates for the diameter of primitive lattice zonotopes in fixed dimensions and identifies conditions under which the maximum diameter in a hypercube is uniquely achieved, revealing growth rates as the hypercube size increases.

## Contribution

It establishes precise asymptotic bounds for the diameter of primitive zonotopes and characterizes the unique maximizers within hypercubes for infinitely many sizes.

## Key findings

- Largest diameter grows like k^{d/(d+1)} for fixed dimension d as k increases.
- Unique maximizers of the diameter are primitive zonotopes for infinitely many k.
- Provides new lower bounds on the maximum diameter of lattice polytopes in hypercubes.

## Abstract

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 $[0,k]^d$.