2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141032We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.16 pagesDifferential Geometry53A07; 53C20; 53C21text