The sharp doubling threshold for approximate convexity

We show for $A,B\subset {\mathbb{R}}^d$ of equal volume and $t\in (0,1\mathrm{/}2]$ that if $\mathrm{\mid }tA+(1-t)B\mathrm{\mid }<(1+t^d)\mathrm{\mid }A\mathrm{\mid }$, then (up to translation) $\mathrm{\mid }\text{co}(A\cup B)\mathrm{\mid }\mathrm{/}\mathrm{\mid }A\mathrm{\mid }$ is bounded. This establishes the sharp threshold for Figalli and Jerison’s quantative stability of the Brunn-Minkowski inequality. We additionally establish a similar sharp threshold for iterated sumsets.