  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 $C^2$-hypersurfaces. We apply these results to prove
$C^{1,\beta}$-convergence of inverse F-curvature flows in the sphere to an
equator in \mathbb{S}^{n+1} for embedded, closed, 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.
