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 conjecture of Figalli and Jerison, we show there universal constants $\\smash{C_n,d_n>0}$ such that for $\\smash{A \\subset \\mathbb{R}^n}$ positive measure, if $\\smash{|\\frac{A+A}{2}\\setminus A| \\le d_n |A|}$, then $\\smash{\\lvert\\operatorname{co}(A)\\setminus C_n |\\frac{A+A}{2}\\setminus A|}$ $\\smash{\\operatorname{co}(A)}$ convex hull $\\smash{A}$.