Evaluating inclusion functionals for random convex hulls

Abstract

The inclusion functional of a random convex set, evaluated at a fixed convex set K, measures the probability that the random convex set contains K. This functional is an analogue of the complement of the distribution function of an ordinary random variable. A methodology is described for evaluating the inclusion functional for the case where the random convex set is generated as the convex hull of n i.i.d. points from a distribution function F in the plane. For general K and F, the inclusion probability is difficult to compute in closed form. The case where K is a straight line segment is examined in detail and, in this situation, a simple answer is given for an interesting class of distributions F.