# On the boundary of almost isoperimetric domains

## TL;DR

This paper demonstrates that nearly optimal isoperimetric domains in Euclidean space have boundaries closely approximating spheres, with refined closeness under curvature bounds, and addresses a question about almost extremal hypersurfaces.

## Contribution

It establishes quantitative boundary closeness of near-isoperimetric domains to spheres and refines this relation with curvature bounds, also answering a specific open question.

## Key findings

- Boundaries of small deficit domains are Hausdorff-close to spheres.
- Refined closeness results under integral curvature bounds.
- Resolved an open question about almost extremal hypersurfaces.

## Abstract

We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Colbois concerning the almost extremal hypersurfaces for Chavel's inequality.