Eventual quasi-linearity of the Minkowski length

Abstract

The Minkowski length of a lattice polytope $P$ is a natural generalization of the lattice diameter of $P$. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in $P$. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates $tP$ of a lattice polytope $P$ behaves polynomially in $t\in\mathbb{N}$. In this paper we prove that for any lattice polytope $P$, the Minkowski length of $tP$ for $t\in\mathbb{N}$ is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of coordinates boxes, degree one polytopes, and dilates of unimodular simplices. In addition, we give a new bound for the Minkowski length of lattice polygons and show that the Minkowski length of a lattice triangle coincides with its lattice diameter.