An extremal property of lattice polygons (Q2313591)

In this nicely written paper (and somehow technical), the authors study convex lattice polygons from a viewpoint of the critical number of vertices that guarantees that the polygon contains at least one point of a given square sublattice. The paper is written in the spirit of Minkowski's convex body theorem which tells us that if a compact set in $\mathbb{R}^{d}$ is symmetric with respect to origin and has volume at least $2^d$, then it contains a point of the integral lattice $\mathbb{Z}^d$. The main results of the paper can be formulated as follows. Theorem A. Given an integer $n\geq 3$, any convex integral polygon with $2n + 3$ vertices contains a point of $(n\mathbb{Z})^{2}$. It is worth emphasizing that the constant $2n + 3$ is optimal, since it is easy to construct a polygon with $2n + 2$ vertices lying in the slab $\{(x_1; x_2) : 0 \leq x_{2} \leq n\}$ and free from points of $(n\mathbb{Z})^2$. Theorem B. Given an integer $n\geq 3$, any convex integral polygon free from points of $(n\mathbb{Z})^2$ has at most $2n + 2$ vertices.