An Alexandrov-Fenchel-type inequality for hypersurfaces in the sphere

TL;DR

This paper establishes a new Alexandrov-Fenchel-type inequality for convex hypersurfaces in the sphere by identifying a monotone quantity along the inverse mean curvature flow, advancing geometric analysis in spherical spaces.

Contribution

It introduces a novel monotone quantity along the inverse mean curvature flow and proves an Alexandrov-Fenchel-type inequality for convex hypersurfaces in the sphere.

Findings

Proved a new inequality for convex hypersurfaces in the sphere.

Identified a monotone quantity along the inverse mean curvature flow.

Extended geometric inequalities to spherical spaces.

Abstract

We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the $n$-dimensional sphere, $n \geq 3$.