Ellipsoids and Matrix valued Valuations

A classification is obtained of Borel measurable, GL(n) covariant, symmetric matrix valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by Lutwak, Yang, and Zhang. A classical concept from mechanics is the Legendre ellipsoid or ellipsoid of inertia Γ2K associated with a convex body K ⊂ R. It can be defined as the unique ellipsoid centered at the center of mass of K such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of K. If we fix a scalar product x · y for x, y ∈ R, Γ2K can be defined by the moment matrix M2(K) of K. This is the n × n matrix with coefficients ∫