Quantitative stability for the Brunn-Minkowski inequality

We prove a quantitative stability result for the Brunn-Minkowski inequality: if |A|=|B|=1, t∈[τ,1−τ] with τ>0, and |tA+(1−t)B|1/n≤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.