Note on the variance of generalized random polygons

We consider a probability model in which the hull of a sample of i.i.d. uniform random points from a convex disc K is formed by the intersection of all translates of another suitable fixed convex disc L that contain the sample. Such an object is called a random L -polygon in K . We assume that both K and L have C^2_+ C + 2 smooth boundaries, and we prove upper bounds on the variance of the number of vertices and missed area of random L -polygons assuming different curvature conditions. We also transfer some of our result to a circumscribed variant of this model.