Sharp quantitative stability of the Brunn-Minkowski inequality

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in ${\mathbb{R}}^n$, we have $\mathrm{\mid }A+B{\mathrm{\mid }}^{1\mathrm{/}n}\ge \mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}n}+\mathrm{\mid }B{\mathrm{\mid }}^{1\mathrm{/}n}$. 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.