Isoperimetric inequality for torsional rigidity in multidimensional domains

We consider the Saint Venant functional $P$ for the torsional rigidity in arbitrary plane and space domains. Our main result is the following sharp estimate: $P\leq(4/n)m$, where $n$ is the dimension of domains and $m$ is the harmonic mean of inertial moments of a domain with respect to coordinate planes. Extremal domains are some ellipsoids. Hence, we obtain a generalization of the isoperimetric inequality, proved by E. Nicolay for the torsional rigidity of simply connected planar domains.