The sharp doubling threshold for approximate convexity

Peter van Hintum, Peter Keevash

We show for $A,B\subset {\mathbb{R}}^d$ of equal volume and $t\in (0,1/2]$ that if $|tA+(1-t)B|<(1+t^d)|A|$, then (up to translation) $|\text{co}(A\cup B)|/|A|$ is bounded. This esta…