The square-root stability conjecture for Brunn–Minkowski

Let $n\in\mathbb{N}$ with $n\geq 2$, let $t\in(0,1/2)$, and let $A,B\subset\mathbb{R}^n$ be measurable sets of equal measure satisfying

$$|tA+(1-t)B|\leq(1+\delta)|A|,$$

where $\delta<d_{n,t}$. Square-root stability conjecture. There exist constants $c_n,d_{n,t}>0$ and a convex set $K$ such that, up to translation, $K\supset A,B$ and

$$|K\setminus A|=|K\setminus B|\leq c_nt^{-1/2}\delta^{1/2}|A|.$$

This conjecture quantifies the expected square-root stability of the Brunn–Minkowski inequality: near equality should force both sets to be close to a common convex set. The source presents it as a major folklore conjecture; its resolution status is not specified.