Some shape functionals for the $k$-Hessian equation

For a non-empty, bounded, open, and convex set of class $C^2$, we consider the Torsional Rigidity associated to the $k$-Hessian operator. We first prove Pólya type lower bound for the $k$-Torsional Rigidity in any dimension; then, in order to investigate optimal sets in the Pólya type inequality, we provide two quantitative estimates.