Norms on $\mathbb{R}^2$ that send integer lattice points on integer I've recently encountered a problem during an oral exam which consisted in finding all norms $N$ on $\mathbb{R}^2$ such that $N(\mathbb{Z}^2)\subseteq \mathbb{N}$
The result is that they are of the form :
$$ N(x,y) = \max_{1\le i\le n} |a_i x + b_i y|$$
With the constraint that the matrix formed by the coefficients $(a_i, b_i)$ is of rank 2.
I'm not asking for a proof as it is quite long, I am asking two things :
1) Can this result be generalized, and what are some interesting applications of this ?
2) Does anyone know any books or document on this notion ?
Thanks in advance.
 A: 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:


*

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

*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
