Some sharp isoperimetric-type inequalities on Riemannian manifolds

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci curvature lower bound $(n-1)k$, the geodesic ball of radius $l$ in the space form of curvature $k$ has the largest area-to-volume ratio. A similar but reversed inequality holds if we replace a lower bound on the cut distance by a lower bound of the mean curvature. As an application we show that $C^2$ isoperimetric domains in standard space forms are balls. Generalized convexity also provides a simple proof of Toponogov theorem. We also prove another isoperimetric inequality involving the extrinsic radius of a domain when the curvature of the ambient space is bounded above. We then extend this inequality in two directions: one involves the higher order mean curvatures, and the other involves the Hausdorff measure of the cut locus.