On the Expected Complexity of Random Convex Hulls

In this paper we present several results on the expected complexity of a convex hull of n points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of n points, chosen uniformly and independently from a disk is O(n1/3), and O(k log n) for the case a convex polygon with k sides. Those results are well known (see [RS63, Ray70, PS85]), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let D be a set of directions in the plane, we define a generalized notion of convexity induced by D, which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the D-convex hull of a set of n points, chosen uniformly and independently from a disk, is O (