Sharp quantitative stability of the planar Brunn–Minkowski inequality

Abstract

We prove a sharp stability result for the Brunn–Minkowski inequality for $A,B\subset {\mathbb{R}}^2$. Assuming that the Brunn–Minkowski deficit $\delta =|A+B{|}^{1/2}/(|A{|}^{1/2}+|B{|}^{1/2})-1$ is sufficiently small in terms of $t=|A{|}^{1/2}/(|A{|}^{1/2}+|B{|}^{1/2})$, there exist homothetic convex sets $K_A\supset A$ and $K_B\supset B$ such that $\frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|}\le C{t}^{-1/2}{\delta }^{1/2}$. The key ingredient is to show for every $ϵ,t>0$, if $\delta$ is sufficiently small then $|\mathrm{co}(A+B)\setminus (A+B)|\le (1+ϵ)(|\mathrm{co}(A)\setminus A|+|\mathrm{co}(B)\setminus B|)$.