Sharp Stability of Brunn-Minkowski for Homothetic Regions

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 $\mathrm{\mid }\frac{A+A}{2}\setminus A\mathrm{\mid }\le d_n\mathrm{\mid }A\mathrm{\mid }$, then $\mathrm{\mid }\mathrm{co}(A)\setminus A\mathrm{\mid }\le C_n\mathrm{\mid }\frac{A+A}{2}\setminus A\mathrm{\mid }$ for $\mathrm{co}(A)$ the convex hull of $A$.