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{Rigidity results, inverse curvature flows and Alexandrov-Fenchel type inequalities in the sphere}

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We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex hypersurfaces in the class C². We apply these results to prove C^{1, β}-convergence of inverse F-curvature flows in the sphere to an equator in the sphere for embedded, closed, orientable, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.