Hypersurfaces with small extrinsic radius or large ${\lambda }_1$ in Euclidean spaces

We prove that hypersurfaces of ${\mathbb{R}}^{n+1}$ 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 $q⩽n$.