A convex lattice polytope is the convex hull of a set of integral points. Vershik conjectured that a polygon chosen uniformly at random from the set of all 1/n scaled convex lattice polygons contained in the unit square will be arbitrarily close to a fixed limit shape, with arbitrarily high probability for n sufficiently large. This conjecture is now a theorem, with three different proofs given by Bárány, by Vershik, and by Sinai. However, the existence of a limit shape for polytopes is open in dimensions greater than 2.
I will discuss this problem and the known results, and I will present recent work, joint with Bárány and Bureaux, in which we show the existence of a limit shape for lattice zonotopes in all dimensions.