Quantitative stability for the Brunn-Minkowski inequality

We prove a quantitative stability result for the Brunn-Minkowski inequality: if $|A|=|B|=1$, $t \in [τ,1-τ]$ with $τ>0$, and $|tA+(1-t)B|^{1/n}\leq 1+δ$ for some small $δ$, then, up to a translation, both $A$ and $B$ are quantitatively close (in terms of $δ$) to a convex set $K$.

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