*Accepted Paper*

**Inserted:** 16 aug 2016

**Last Updated:** 16 aug 2016

**Journal:** Chin. Ann. Math.

**Year:** 2016

**Abstract:**

We prove a quantitative stability result for the
Brunn-Minkowski inequality on sets of equal volume: if $

A

=

B

>0$
and $

A+B

^{1/n}=(2+\delta)

A

^{1/n}$ for some small $\delta$,
then, up to a translation, both $A$ and $B$ are close (in terms of $\delta$)
to a convex set $\K$. Although this result was already proved in our previous paper \cite{fjAB}
even for sets of different volume,
we provide here a more elementary proof that we believe has its own interest.
Also, in terms of the stability exponent, this result provides a stronger estimate than the result in \cite{fjAB}.

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