  Positive solutions of homogeneous Dirichlet boundary value problems or
initial-value problems for certain elliptic or parabolic equations must be
radially symmetric and monotone in the radial direction if just one of their
level surfaces is parallel to the boundary of the domain. Here, for the
elliptic case, we prove the stability counterpart of that result. In fact, we
show that if the solution is almost constant on a surface at a fixed distance
from the boundary, then the domain is almost radially symmetric, in the sense
that is contained in and contains two concentric balls $B_{r_e}$ and $B_{r_i}$,
with the difference $r_e-r_i$ (linearly) controlled by a suitable norm of the
deviation of the solution from a constant. The proof relies on and enhances
arguments developed in a paper by Aftalion, Busca and Reichel.
