In this talk, I will introduces a new notion of convexity on the unit sphere, called horo-convexity, inspired by its analogue in hyperbolic space. For horo-convex hypersurfaces, we prove the smooth convergence of the Guan–Li inverse curvature flow and, as a consequence, establish the full set of quermassintegral inequalities on the sphere. The talk will briefly outline the definition, the flow approach, and the main geometric results. This talk is based on joint work with Julian Scheuer
