Quantitative stability for the Brunn-Minkowski inequality

TL;DR

This paper establishes a quantitative stability result for the Brunn-Minkowski inequality, showing that near equality implies the sets are close to convex sets, up to translation.

Contribution

It provides a new quantitative stability theorem for the Brunn-Minkowski inequality with explicit bounds relating set closeness to near equality.

Findings

Sets are close to convex sets when the inequality nearly holds.

Quantitative bounds depend on the deviation from equality.

Results hold for sets of equal measure with specified parameters.

Abstract

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