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Article
Eventual Quasi-Linearity of The Minkowski Length
European Journal of Combinatorics
  • Ivan Soprunov, Cleveland State University
  • Jenya Soprunova, Kent State University
Document Type
Article
Publication Date
11-1-2016
Disciplines
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
Citation Information
Ivan Soprunov and Jenya Soprunova. "Eventual Quasi-Linearity of The Minkowski Length" European Journal of Combinatorics Vol. 58 (2016) p. 107 - 117
Available at: http://works.bepress.com/ivan-soprunov/6/