On the Computer Generation Random Convex Hulls

The convex hull of X,,. . ., X., a sample of independent identically distributed Rd-valued random vectors with density f is called a random convex hull with parameters f and II. In this paper, we give an a orithm for the computer generation of random convex hulls when f is radial, i.e. when j(x) = g(ll $ l)Tor some function g. Then we look at the average time E(r) of the algorithm under a convenient computatibnal model. We consider only d = 2. We show that for any f, our &orithm takes average time fi(log n). This lower bound is achieved for all radial densities with a polynomially decreasing tail. For the radial densities with an exponentially decreasing tail, we $ha,w that E(T) = O(log3’*n). Finally, for the uniform density on the unit circle, we have E( 7J = O(,“l lo~?n). This rate is also shown to be optimal for this density.