  he celebrated formula of Schlafli relates the variation of the dihedral
angles of a smooth family of polyhedra in a space form and the variation of
volume. We give a smooth analogue of this classical formula -- our result
relates the variation of the volume bounded by a hypersurface moving in a
general Einstein manifold and the integral of the variation of the mean
curvature. The argument is direct, and the classical polyhedral result (as well
as results for Lorenzian space forms) is an easy corollary. We extend it to
variations of the metric in a Riemannian Einstein manifold with boundary. We
apply our results to extend the classical Euclidean inequalities of Aleksandrov
to other 3-dimensional constant curvature spaces. We also obtain rigidity
results for Ricci-flat manifolds with umbilic boundaries and existence results
for foliations of Einstein manifolds by hypersurfaces.
