Alexandrov-Fenchel type inequalities for convex hypersurfaces in hyperbolic space and in sphere
Abstract
In this paper, firstly, inspired by Nat\'ario's recent work Na, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space n+1 and in the sphere n+1. We also get the rigidity in the spherical case. Secondly, we use the inverse mean curvature flow in sphere gerh,Mak-Sch to prove an optimal Sobolev type inequality for closed convex hypersurfaces in the sphere.
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