  We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$
where $\Omega$ is a bounded domain and $f$ is a convex function. Under general
assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a
minimizer in $W_0^{1,1}(\Omega)$ depending only on the distance from the
boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on
$f$, 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.
