Eventual quasi-linearity of the Minkowski length

TL;DR

This paper proves that the Minkowski length of dilated lattice polytopes eventually follows a quasi-polynomial pattern with linear components, extending understanding of lattice polytope geometry.

Contribution

It establishes the eventual quasi-linearity of Minkowski length for dilated lattice polytopes and provides explicit formulas and bounds for specific cases.

Findings

Minkowski length of $tP$ is eventually a quasi-polynomial with linear parts.

Explicit formulas for Minkowski length of boxes, degree one polytopes, and unimodular simplices.

New bounds for Minkowski length of lattice polygons and triangles.

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.