In this note we prove that the ball is a maximiser of some Schatten
p-norms of the Riesz potential operators among all domains of a given measure in
R
d
. In particular, the result is valid for the polyharmonic Newton potential operator,
which is related to a nonlocal boundary value problem for the poly-Laplacian
extending the one considered by M. Kac in the case of the Laplacian, so we obtain
and isoperimetric inequalities for its eigenvalues as well, namely, analogues of
Rayleigh-Faber-Krahn and Hong-Krahn-Szeg¨o inequalities.