Covariance matrices and valuations (Q394811)

The moment matrix of a function \(f\) of real variables \(x_1,\dots,x_n\) is the matrix, whose \((i,j)\)-th entry is the integral of \(x_i x_j f\) over \(\mathbb{R}^n\). The moment matrix can be regarded as a matrix-valued valuation on the space of functions with finite second moments: recall that a valuation on a space of functions \(L\) is a function \(Z\) on \(L\) such that \(Z(f)+Z(g)=Z(\max(f,g)) +Z(\min(f,g))\) for all \(f\) and \(g\) in \(L\).NEWLINENEWLINEThe author proves that the moment matrix is the unique continuous homogeneous valuation on the space of functions with finite second moments, which is \(\mathrm{SL}(n)\) covariant, i.e. \(Z(f) = M Z(f \circ M) M^T\) for every function \(f\) on \(\mathbb{R}^n\) and every \(M\) in \(\mathrm{SL}(n)\). The author also classifies all continuous \(\mathrm{SL}(n)\) covariant valuations with no assumption of homogeneity.NEWLINENEWLINEA similar Hadwiger-type characterization of the Fisher information matrix was obtained by the author in [Adv. Math. 226, No. 3, 2700--2711 (2011; Zbl 1274.62064)].