There has been quite a lot of work done recently in various generalized models of random polytopes in convex bodies. One such model is when one takes n independent identically distributed uniform random points from a suitable convex body and considers the intersection of all congruent closed balls that contain the points. The resulting intersection is called a random ball-polytope. One is usually interested in the behaviour of both the metric properties of such objects, such as volume, surface area,intrinsic volumes, and also the combinatorial structure, like number of vertices, facets, etc. As in the case of classical random polytopes, this behaviour heavily depends on the smoothness of the boundary of the mother convex body.

In this talk we discuss the behaviour of the vertex number and area of random disc-polygons, which is the two-dimensional case of the above model. Our aim is to prove series expansions for the expectation of the vertex number and area of random disc-polygons depending on the degree of smoothness of the boundary of the convex disc.

Joint work with N. Montenegro (University of Szeged, Hungary).