  We give a curvature dependent lower bound for the filling radius of all
closed Riemannian manifolds as well as an upper one for manifolds which are the
total space of a Riemannian submersion. The latter applies also to the case of
submetries. We also see that the reach (in the sense of Federer) of the image
of the Kuratowski embedding vanishes, and we finish by giving some inequalities
involving the k-intermediate filling radius.
