Document Type

Article

Publication Date

11-1-2016

Publication Title

European Journal of Combinatorics

Abstract

The Minkowski length of a lattice polytope PP is a natural generalization of the lattice diameter of PP. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in PP. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates tPtP of a lattice polytope PP behaves polynomially in t∈Nt∈N. In this paper we prove that for any lattice polytope PP, the Minkowski length of tPtP for t∈Nt∈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.

Comments

The first author is partially supported by NSA Grant H98230-13-1-0279.

DOI

10.1016/j.ejc.2016.05.009

Version

Postprint

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Volume

58

Included in

Mathematics Commons

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