# Quantitative stability of the Brunn-Minkowski inequality for sets of equal volume

created by figalli on 16 Aug 2016

[BibTeX]

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}.