Abstract

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(n 1=3 ). Applying the same technique to the case where the points are chosen from a convex polygon with k sides, is O(k log n). 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 i n 1=3 + p nff(D) j , where ff(D) is the largest angle between two consecutive vectors in D. This result extends the known bou...

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