Sharp quantitative stability of the planar Brunn-Minkowski inequality

TL;DR

This paper establishes a precise stability estimate for the planar Brunn-Minkowski inequality, showing that small deficits imply the sets are close to convex sets, with explicit bounds.

Contribution

It provides the first sharp quantitative stability result for the planar Brunn-Minkowski inequality, linking small deficits to proximity to convex sets.

Findings

Small Brunn-Minkowski deficit implies sets are close to convex sets.

Quantitative bounds relate deficit size to set proximity.

Key inequality connects the convex hulls of sumsets with original sets.

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 $\delta=|A+B|^{\frac{1}{2}}/(|A|^\frac12+|B|^\frac12)-1$ 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 $|co(A+B)\setminus (A+B)|\le (1+\epsilon)(|co(A)\setminus A|+|co(B)\setminus B|)$.