The quermassintegral inequalities for horo-convex domains in the sphere

We study a new notion of convexity for subsets of the unit sphere, which closely resembles the horo-convexity for subsets of the hyperbolic space. We call this notion, accordingly, horo-convexity. For horo-convex hypersurfaces of the unit sphere, we prove the smooth convergence of the classical Guan/Li flow of inverse type and use this result to prove the full set of quermassintegral inequalities for horo-convex hypersurfaces of the unit sphere.