Norms on $\mathbb{R}^2$ that send integer lattice points on integer

A similar result holds in all dimensions.

Theorem. Suppose that $N$ is a norm on ${\mathbb R}^n$ which takes integer values on ${\mathbb Z}^n$. Then there are finitely many linear functions $\ell_i: {\mathbb R}^n\to {\mathbb R}, i\in I,$ with integer coefficients (i.e. integer linear functions), such that $$ N(x)= \max_{i\in I} \ell_i(x). $$ I do not have a reference for this and a proof is a bit long. The main step is the following:

Lemma. Let $B=N^{-1}([0,1])$ denote the unit ball of the norm $N$; let $S$ denote the boundary of $B$. For every $x\in S_{{\mathbb Q}}=S \cap {\mathbb Q}^n$, there exists an affine function $F: {\mathbb R}^n\to {\mathbb R}$ such that:

• $F(x)=1$.
• $F|_B \le N|_B$.
• The linear part $\ell$ of $F$ is an integer linear function.
• $\|\ell\|\le C$, where $C$ is a constant depending only on the norm $N$.

Once this lemma is proven, one uses density of $S_{{\mathbb Q}}$ in $S$ and the fact that there are only finitely many linear functions $\ell$ which can appear in this lemma, to conclude the proof of the theorem.

Edit. Regarding applications:

• Integer-valued norms (and the associated polyhedra, their unit balls) appear as important topological invariants. See for instance:

  1. Thurston norm. (One can find more references just by googling "Thurston norm".)

  2. Alexander norm, in this paper.

• The existence of a discrete norm characterizes free abelian groups among all abelian groups:

Juris Steprāns, A Characterization of Free Abelian Groups, Proceedings of the American Mathematical Society, Vol. 93, No. 2 (1985), pp. 347-349