In the first part of the talk we replace the unit disc in the process by regular simplices (in any dimension) and by regular polygons (in the plane). Asymptotic results are given for the speed of the process, and we examine the limit distribution of the center in the case of simplices.
In the second part of the talk we consider another model to obtain random disc-polygons: we fix a spindle convex disc $S$ in the plane, we choose $n$ independent random point from $S$ according to the uniform distribution, and we define $S_n$ as the spindle convex hull of the chosen points. We show asymptotic results for the expectation of the number of the vertices of $S_n$, if $S$ has smooth enough boundary, or if $S$ is a disc-polygon.
The talk is based on joint work with G. Ambrus, F. Fodor and P. Kevei.