THE EXPECTED VALUE OF SOME FUNCTIONS OF THE CONVEX-HULL OF A RANDOM SET OF POINTS SAMPLED IN RD

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