# Weighted Alexandrov-Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang and Wu

## TL;DR

This paper investigates weighted Alexandrov-Fenchel inequalities in hyperbolic space, proving a similar inequality to a conjecture and providing a counterexample in a specific case.

## Contribution

It proves a near version of a conjectured inequality for horospherically convex hypersurfaces and presents a counterexample in three dimensions.

## Key findings

- Proved a similar inequality to the conjecture for certain hypersurfaces.
- Provided a counterexample to the conjecture in three-dimensional space.
- Enhanced understanding of geometric inequalities in hyperbolic space.

## Abstract

We consider a conjecture made by Ge, Wang and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the $k^{\mathrm{th}}$ mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when $k$ is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.