Symmetry of minimizers with a level surface parallel to the boundary

We consider an integral functional representing torsional rigidity with a general convex lagrangean. G. Crasta has shown that if the functional admits a minimizer depending only on the distance from the boundary of the domain, then the domain must be a ball. With some restrictions on the lagrangean, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.