  We show that the modern theory of fully nonlinear operators had been started
by the skew symmetry of minors in cooperation with the symmetry of symmetric
functions. The paper presents some consequences of this interaction for the
m-Hessian operators. One of them is a setting of the isoperimetric variational
problem for Hessian integral. The m-admissible minimizer is found, what brings
out a new simple proof of the well known Poincare - type inequalities for
Hessian integrals. Also a new set of inequalities, generated by a special
finite set of functions, is found.
