# A stability result for the cube edge isoperimetric inequality

## TL;DR

This paper establishes a stability version of the cube edge isoperimetric inequality, showing that near-minimal boundary sets are close to unions of disjoint cubes, extending previous results and offering a dimension-free perspective.

## Contribution

It provides a new stability result for the cube edge isoperimetric inequality that is independent of dimension, extending Ellis's work and relating to Friedgut's junta theorem.

## Key findings

- Sets with near-minimal boundary are close to unions of disjoint cubes
- The stability result is dimension-free and applies broadly
- Extends previous stability results by Ellis

## Abstract

We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon $-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.