An identity relating moments of functionals of convex hulls

Abstract

Denote by Kn the convex hull of n independent random points distributed uniformly in a convex body K in ℝd, by Vn the volume of Kn, by Dn the volume of K\Kn, and by Nn the number of vertices of Kn. A well-known identity due to Efron relates the expected volume EDn - and thus EVn - to the expected number ENn+1. This identity is extended from expected values to higher moments. The planar case of the arising identity for the variances provides in a simple way the corrected version of a central limit theorem for Dn by Cabo and Groeneboom (K being a convex polygon) and an improvement of a central limit theorem for Dn by Hsing (K being a circular disk). Estimates of varDn (K being a two-dimensional smooth convex body) and varNn (K being a d-dimensional smooth convex body, d ≥ 4) are obtained. The identity for moments of arbitrary order shows that the distribution of Nn determines EVn-1, EVn-22,..., EV d+1n-d-1. Reversely it is proved that these n-d-1 moments determine the distribution of Nn entirely. The resulting formula for the probability that Nn = k (k = d + 1,...,n) appears to be new for k ≥ d + 2 and yields an answer to a question raised by Baryshnikov. For k = d + 1 the formula reduces to an identity which has been repeatedly pointed out.