# Round spheres are Hausdorff stable under small perturbation of entropy

## TL;DR

This paper proves that closed hypersurfaces with entropy close to that of a sphere are geometrically close in Hausdorff distance, extending previous results to higher dimensions.

## Contribution

It generalizes the Hausdorff stability of round spheres under small entropy perturbations to higher-dimensional hypersurfaces.

## Key findings

- Hypersurfaces with near-sphere entropy are Hausdorff close to a sphere.
- The result extends previous two-dimensional stability results to higher dimensions.
- Provides a quantitative link between entropy and geometric closeness.

## Abstract

We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.