On two symmetries in the theory of $m$-Hessian operators

Abstract

The modern theory of fully nonlinear operators had been inspired by the skew symmetry of minors in cooperation with the symmetry of symmetric functions. We present some consequences of this interaction for $m$-Hessian operators. One of them is setting of the isoperimetric variational problem for Hessian integrals. The $m$-admissible minimizer is found that allows a new simple proof of the well-known Poincaré-type inequalities for Hessian integrals. Also a new set of inequalities, generated by a special finite set of functions, is presented.