On norms taking integer values on the integer lattice

TL;DR

This paper provides a new proof of Thurston's theorem, showing that seminorms on real space that take integer values on the integer lattice have polyhedral unit balls defined by finitely many integer coefficient inequalities.

Contribution

The paper introduces a novel proof of Thurston's theorem regarding the polyhedral structure of seminorms with integer lattice values.

Findings

Seminorm unit balls are polyhedra with finitely many inequalities.

The inequalities defining the polyhedron have integer coefficients.

The proof offers new insights into the structure of such seminorms.

Abstract

We present a new proof of Thurston's theorem that the unit ball of a seminorm on $\mathbf{R}^d$ taking integer values on $\mathbf{Z}^d$ is a polyhedra defined by finitely many inequalities with integer coefficients.