# Sharp Stability of Brunn-Minkowski for Homothetic Regions

## TL;DR

This paper establishes a precise stability result for the Brunn-Minkowski inequality, showing that sets nearly attaining equality are close to convex sets, thereby confirming a conjecture by Figalli and Jerison.

## Contribution

It proves a sharp stability theorem for homothetic sets in the Brunn-Minkowski inequality, resolving a conjecture and providing universal constants for the stability estimate.

## Key findings

- Sets with near-equality are close to convex sets.
- Confirmed a conjecture of Figalli and Jerison.
- Provided explicit universal constants for stability.

## Abstract

We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn-Minkowski inequality are to being convex. In particular, resolving a conjecture of Figalli and Jerison, we show there are universal constants $C_n,d_n>0$ such that for $A \subset \mathbb{R}^n$ of positive measure, if $|\frac{A+A}{2}\setminus A| \le d_n |A|$, then $|\operatorname{co}(A)\setminus A| \le C_n |\frac{A+A}{2}\setminus A|$ for $\operatorname{co}(A)$ the convex hull of $A$.