Quantitative stability for the Brunn-Minkowski inequality

We prove a quantitative stability result for the Brunn-Minkowski inequality: if $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }B\mathrm{\mid }=1$, $t\in [\tau ,1-\tau ]$ with $\tau >0$, and $|tA+(1-t)B|^{1/n}\leq 1+\delta$forsomesmall$\delta$,then,uptoatranslation,both$A$and$B$arequantitativelyclose(intermsof$\delta$)toaconvexset$K$.