The quermassintegral inequalities for horo-convex domains in the sphere

TL;DR

This paper introduces a new notion of convexity called horo-convexity for subsets of the unit sphere, and proves quermassintegral inequalities for these hypersurfaces using flow methods.

Contribution

It defines horo-convexity on the sphere and establishes the full set of quermassintegral inequalities for these hypersurfaces, extending convex geometric inequalities.

Findings

Proves smooth convergence of the Guan/Li flow for horo-convex hypersurfaces.

Establishes quermassintegral inequalities for horo-convex domains.

Introduces a new convexity notion for spherical subsets.

Abstract

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.