INEQUALITIES OF ALEXANDROV-FENCHEL TYPE FOR CONVEX HYPERSURFACES IN HYPERBOLIC SPACE AND IN THE SPHERE

By applying the unit normal flow to well-known inequalities in hyperbolic space Hn+1 and in the sphere Sn+1, we derive some new inequalities of Alexandrov-Fenchel type for closed convex hypersurfaces in these spaces. We also use the inverse mean curvature flow in the sphere to prove an optimal Sobolev-type inequality for closed convex hypersurfaces in the sphere.