Edit: Fix a lattice $N = \mathbb{Z}^n$ and let $N\_{\mathbb{R}} = N \otimes \mathbb{R}$.
Let $C(S)$ be a strongly convex rational cone in $N\_{\mathbb{R}}$ generated by a finite set $S \subset N$, with following two properties:


* No $v \in S$ is in the interior of $C(S)$.
* $S$ generates $N$ as a $\mathbb{Z}$-module.


I have a vector $v \in N$, which has the property:



> 
> $v$ is NOT a nonnegative $\mathbb{Z}$-linear combination of vectors in $S$. 
> 
> 
> 


Does it imply that $v \notin C(S)$? If not, is there any simple low-dimensional counterexample?



A question about rational convex cone
Counterexample: lattice is $\mathbb Z^2$, $S = \{(1,0),(1,1),(1,3)\}$, $v=(1,2)$.


