  We prove that hypersurfaces of $\R^{n+1}$ which are almost extremal for the
Reilly inequality on $\lambda_1$ and have $L^p$-bounded mean curvature ($p>n$)
are Hausdorff close to a sphere, have almost constant mean curvature and have a
spectrum which asymptotically contains the spectrum of the sphere. We prove the
same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We
also prove that when a supplementary $L^q$ bound on the second fundamental is
assumed, the almost extremal manifolds are Lipschitz close to a sphere when
$q>n$, but not necessarily diffeomorphic to a sphere when $q\leqslant n$.
