Sharp quantitative stability of the Brunn-Minkowski inequality

Abstract

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in ${\mathbb{R}}^n$, we have $|A+B{|}^{1/n}\ge |A{|}^{1/n}+|B{|}^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in ${\mathbb{R}}^d$. The stability of this statement is a well-known problem that has attracted much attention in recent years. This paper gives a conclusive answer by proving the sharp stability result for the Brunn-Minkowski inequality on arbitrary sets.

54 pages