The volume of an isotropic random parallelotope

Abstract

The p-content of the p-parallelotope ∇ p, n determined by p independent isotropic random points z 1, …, zp in ℝ n (1 < p ≦ n) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points, enabling calculation of the actual distribution as well as the moments of ∇ p, n . This is done explicitly in several cases. The results also have interest in multivariate statistics.