Alexandrov-Fenchel type inequalities in the sphere

TL;DR

This paper explores Alexandrov-Fenchel inequalities on the sphere by employing two types of geometric flows to analyze relations between quermassintegrals for convex hypersurfaces.

Contribution

It introduces a novel approach using two types of flows to establish inequalities between quermassintegrals on the sphere.

Findings

Established relations between quermassintegrals using geometric flows.

Extended Alexandrov-Fenchel inequalities to convex hypersurfaces in spherical space.

Provided new techniques for analyzing convex hypersurfaces in differential geometry.

Abstract

In this paper, we attempt to use two types of flows to study the relations between quermassintegrals $\mathcal{A}_k$ (see Definition 1.1), which correspond to the Alexandrov-Fenchel inequalities for closed convex $C^2$-hypersurfaces in $\mathbb{S}_+^{n+1}.$