Sharp stability of Brunn–Minkowski for homothetic regions

Abstract

We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn–Minkowski inequality are to being convex. In particular, resolving a conjecture of Figalli and Jerison, we show there are universal constants $C_n,d_n>0$ such that for $A\subset {\mathbb{R}}^n$ of positive measure, if $|\frac{A+A}{2}\setminus A|\le d_n|A|$, then $|\mathrm{co}(A)\setminus A|\le C_n|\frac{A+A}{2}\setminus A|$ for $\mathrm{co}(A)$ the convex hull of $A$.