Finite convex-hull width does not imply a Jordan curve is a quasicircle, but boundary width below cosh^{-1}(sqrt(2)) does.
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Quasicircles and width of Jordan curves in $\mathbb{CP}^1$
Finite convex-hull width does not imply a Jordan curve is a quasicircle, but boundary width below cosh^{-1}(sqrt(2)) does.