Simplicial variances, potentials and Mahalanobis distances

The average squared volume of simplices formed by k independent copies from the same probability measure μ on Rd defines an integral measure of dispersion ψk(μ), which is a concave functional of μ after suitable normalization. When k=1 it corresponds to tr(Σμ) and when k=d we obtain the usual generalized variance det(Σμ), with Σμ the covariance matrix of μ. The dispersion ψk(μ) generates a notion of simplicial potential at any x∈Rd, dependent on μ. We show that this simplicial potential is a quadratic convex function of x, with minimum value at the mean aμ for μ, and that the potential at aμ defines a central measure of scatter similar to ψk(μ), thereby generalizing results by Wilks (1960) and van der Vaart (1965) for the generalized variance. Simplicial potentials define generalized Mahalanobis distances, expressed as weighted sums of such distances in every k-margin, and we show that the matrix involved in the generalized distance is a particular generalized inverse of Σμ, constructed from its characteristic polynomial, when k=rank(Σμ). Finally, we show how simplicial potentials can be used to define simplicial distances between two distributions, depending on their means and covariances, with interesting features when the distributions are close to singularity.