Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose coordinates are coprime) directing the two edges generate the whole lattice, or in other words, the associated toric surface is smooth.
Take a lattice point $M$ in the interior of $P$. Is it always possible to find a boundary lattice point $B$ of $P$ which is at integer length $1$ of $M$, i.e. such that the vector $MB$ is primitive? Is there also a toric interpretation of this property?
If one removes the smoothness assumption, there are counter-examples. However, the answer is yes for many simple polygons (e.g. for the ones associated to $\mathbb{C}P^2$, the Hirzebruch surfaces, and some of their blow-ups).
Edit:
We were just given a counter-example by Christian Haase. Take the vectors v from the columns of
1 2 1 1 0 -1
0 1 1 2 1 1
and consider the zonotope of the segments [-v,v]. Then the origin cannot be joined to its boundary with a primitive vector.
Let me refine the question by assuming moreover that the interior polygon $Q$ of $P$ is divisible by $2$ (i.e. its image by an homothetie of center one of its corners and rate $1/2$ is still a lattice polygon). Translate $P$ so that one of the corner of $Q$ is at the origin. Assume now that the coordinates of $M$ are both even. With all these assumptions, is it now possible to find a boundary point $B$ with the needed property?
