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In this talk we discuss a discrete version of the Minkowski Convex Body Theorem for convex integer polygons on the plane. The theorem states that given a sublattice of the integer lattice, any such polygon with sufficiently many vertices contains at least one point of the sublattice. An explicit formula for the critical number of vertices is provided. The convexity is essential, but we do not need the polygon to be symmetric. Unlike the classical Minkowski's theorem, this statement does not generalise to higher dimensions. The proof can be reduced to estimating the number of vertices of polygons subject to certain geometric constraints. To achieve this, we develop tools that allow to translate the the constraints into nonlinear inequalities in integers.
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