## Prof. Dr. Henry Wynn

*The average squared volume of simplices formed by k independent copies from the same probability measure μ on ℝ^d defines an integral measure of dispersion ψ_k(μ), which is a concave functional of μ after suitable normalisation. When k=1 it corresponds to trace(Σ_μ) and when k=d we obtain the usual generalised variance det( Σ_μ), with Σ_μ the covariance matrix of μ. The dispersion ψ_k(μ) generates a notion of simplicial potential at any x in ℝ^d, dependent on μ. We show that this simplicial potential is a quadratic convex function of x, with minimum value at the mean a_μ for μ, and that the potential at a_μ defines a central measure of scatter similar to ψ_k(μ), thereby generalising results by Wilks (1960) and Van der Waart (1965) for the generalised variance. Simplicial potentials define generalised Mahalanobis distances, expressed as weighted sums of such distances in every k-margin, and we show that the matrix involved in the generalised distance is a particular generalised inverse of Σ_μ, constructed from its characteristic polynomial, when k=rank(Σ_μ). 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.*

Datum: **24.11.2018**, Raum: **G03-106**, Zeit: **17:00**