Sharp stability of the Brunn-Minkowski inequality via optimal mass transportation

TL;DR

This paper proves a sharp stability result for the Brunn-Minkowski inequality using optimal mass transportation, showing how sets close to equality are near convex and homothetic, with a new proof for the quadratic stability conjecture.

Contribution

It provides an alternative proof of the quadratic stability in the Brunn-Minkowski inequality by combining linear stability with optimal transportation techniques.

Findings

Established a quadratic stability result for the Brunn-Minkowski inequality.

Connected stability to optimal transportation methods.

Provided a new proof approach for the stability conjecture.

Abstract

The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\mathbb{R}^d$, states that $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in $\mathbb{R}^d$. The concept of stability in this context concerns how, when approaching equality, sets $A$ and $B$ are close to homothetic convex sets. In a recent breakthrough [FvHT23], the authors of this paper proved the following folklore conjectures on the sharp stability for the Brunn-Minkowski inequality: (1) A linear stability result concerning the distance from $A$ and $B$ to their respective convex hulls. (2) A quadratic stability result concerning the distance from $A$ and $B$ to their common convex hull. As announced in [FvHT23], in the present paper, we leverage (1) in conjunction with a novel optimal transportation approach to offer an alternative proof for (2).