  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.
