A canonical decomposition of the probability measure of sets of isotropic random points in n

The probability measure of X = (x0,..., xr), where x0,..., xr are independent isotropic random points in n (1 <= r <= n - 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of ([omega], ), the probability measure of p, and the probability measure of Y* = (y0*,..., yr*). Here [omega] is the r-subspace parallel to the r-flat [eta] determined by X, is a unit vector in [omega][perpendicular] with 'initial' point at the origin [[omega][perpendicular] is the (n - r)-subspace orthocomplementary to [omega]], p is the norm of the vector z from the origin to the orthogonal projection of the origin on [eta], and yi* = (xi - z)/[alpha](p2), where [alpha] is a scale factor determined by p. The probability measure for [omega] is the unique probability measure on the Grassmann manifold of r-subspaces in n invariant under the group of rotations in n, while the conditional probability measure of given [omega] is uniform on the boundary of the unit (n - r)-ball in [omega][perpendicular] with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x0,..., xr} for 1 <= r <= n.