On means of determinants of matrix projections

Abstract

We establish determinantal counterparts of classical integral-geometric representations of quermassintegrals of convex bodies in the case of mixed discriminants of positive semidefinite matrices and the identity matrix. In particular, we derive an analogue of the Cauchy-Kubota formulae for those mixed discriminants. This note is inspired by a result by Barvinok (1997), in which the average of the determinants of the projections of a positive semidefinite matrix onto (n-1)-linear subspaces is proven to be equal to a matrix analogue of the surface area of a convex body. Further, the one-to-one relation between positive semidefinite matrices and centered ellipsoids allows us to provide this notion of projection of a matrix with a geometrical insight.