Hyperbolic Alexandrov-Fenchel quermassintegral inequalities I

TL;DR

This paper establishes a new hyperbolic Alexandrov-Fenchel inequality for horospherical convex hypersurfaces in hyperbolic space, relating integral curvature quantities to surface area, with equality characterizing geodesic spheres.

Contribution

It proves a novel geometric inequality in hyperbolic space involving the fourth elementary symmetric function, extending Alexandrov-Fenchel inequalities to this setting.

Findings

Proves a new inequality for horospherical convex hypersurfaces in hyperbolic space.

Characterizes equality cases as geodesic spheres.

Extends classical convex geometric inequalities to hyperbolic geometry.

Abstract

In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_\Sigma \s_4 d \mu\ge \ds\vs C_{n-1}^4\omega_{n-1}\left\{\left(\frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 12 + \left(\frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 12\frac {n-5}{n-1}} \right\}^2, \end{array}\] provided that $\Sigma$ is a horospherical convex hypersurface. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\H^n$.