  We study a notion of "width" for Jordan curves in $\mathbb{CP}^1$, paying
special attention to the class of quasicircles. The width of a Jordan curve is
defined in terms of the geometry of its convex hull in hyperbolic three-space.
A similar invariant in the setting of anti de Sitter geometry was used by
Bonsante-Schlenker to characterize quasicircles amongst a larger class of
Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS
setting, we show that there are Jordan curves of bounded width which fail to be
quasicircles. However, we show that Jordan curves with small width are
quasicircles.
