The expected value of some functions of the convex hull of a random set of points sampled inR d

Abstract

This paper presents formulas and asymptotic expansions for the expected number of vertices and the expected volume of the convex hull of a sample ofn points taken from the uniform distribution on ad-dimensional ball. It is shown that the expected number of vertices is asymptotically proportional ton (d−1)/(d+1), which generalizes Renyi and Sulanke’s asymptotic raten (1/3) ford=2 and agrees with Raynaud’s asymptotic raten (d−1)/(d+1) for the expected number of facets, as it should be, by Barany’s result that the expected number ofs-dimensional faces has order of magnitude independent ofs. Our formulas agree with the ones Efron obtained ford=2 and 3 under more general distributions. An application is given to the estimation of the probability content of an unknown convex subset ofR d .