Volumes and intrinsic diameters of hypersurfaces

We estimate the volume and the intrinsic diameter of a hypersurface MM with geometric information of a hypersurface which is parallel to MM at distance TT. It can be applied to the Riemannian Penrose inequality to obtain a lower bound of the total mass of a spacetime. Also it can be used to obtain upper bounds of the volume and the intrinsic diameter of the celestial rr-sphere without a lower bound of the sectional curvature. We extend our results to metric-measure spaces by using the Bakry–Emery Ricci tensor.