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 $\delta =|A+B{|}^{\frac{1}{2}}/(|A{|}^{\frac{1}{2}}+|B{|}^{\frac{1}{2}})-1$ is sufficiently small in terms of $t=|A{|}^{\frac{1}{2}}/(|A{|}^{\frac{1}{2}}+|B{|}^{\frac{1}{2}})$, there exist homothetic convex sets $K_A\supset A$ and $K_B\supset B$ such that $\frac{|K_A\setminus A|}{|A|}+\frac{|K_B\setminus B|}{|B|}\le C{t}^{-\frac{1}{2}}{\delta }^{\frac{1}{2}}$. The key ingredient is to show for every $\epsilon >0$, if $\delta$ is sufficiently small then $|co(A+B)\setminus (A+B)|\le (1+\epsilon )(|co(A)\setminus A|+|co(B)\setminus B|)$.

29 pages, 19 figures. Comments welcome