Title: “Limit shape of random polygons and lattice polygons: a survey”

Abstract. Assume K is a convex body in the plane and X is a (large) finite

subset of K. How many convex polygons are there whose vertices belong to X? Is there a typical shape of such polygons?  In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester’s famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers.