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

AbstractThe probability measure of X = (x0,…, xr), where x0,…, xr are independent isotropic random points in Rn (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 (ω, ẑ), the probability measure of p, and the probability measure of Y∗ = (y0∗,…, yr∗). Here ω is the r-subspace parallel to the r-flat η determined by X, ẑ is a unit vector in ω⊥ with ‘initial’ point at the origin [ω⊥ is the (n − r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and yi∗ = (xi − z)α(p2), where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in Rn invariant under the group of rotations in Rn, while the conditional probability measure of ẑ given ω is uniform on the boundary of the unit (n − r)-ball in ω⊥ 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