Minkowski length of 3D lattice polytopes

TL;DR

This paper introduces a polynomial-time algorithm to compute the Minkowski length of 3D lattice polytopes and explores properties of polytopes with Minkowski length 1, extending known 2D results to three dimensions.

Contribution

It provides the first polynomial-time algorithm for calculating Minkowski length in 3D and analyzes the structure of polytopes with Minkowski length 1, including bounds on interior lattice points.

Findings

Polynomial-time algorithm for L(P) in 3D

Bound on interior lattice points for Minkowski length 1 polytopes

Extension of 2D lattice polytope results to 3D

Abstract

We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum of L = L(P) lattice polytopes Qi, each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q1,..., QL is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case.