# Weighted geometric inequalities for hypersurfaces in sub-static manifolds

## TL;DR

This paper establishes new weighted geometric inequalities for convex hypersurfaces in Euclidean space and certain manifolds, linking curvature, area, and volume, with applications to specific geometric configurations.

## Contribution

It introduces two novel weighted geometric inequalities for hypersurfaces, extending classical results to more general settings including specific manifolds.

## Key findings

- Proved weighted inequalities involving area, volume, and mean curvature.
- Extended inequalities to sphere and AdS-Reissner-Nordström manifold.
- Provided an example of a convex surface with unique inertia-to-area ratio.

## Abstract

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curvature and the area of the hypersurface. Versions of the first inequality for the sphere and for the adS-Reissner-Nordstr\"om manifold are proven. We end with an example of a convex surface for which the ratio between the polar moment of inertia and the square of the area is less than that of the round sphere.