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Sullivan showed that there exists $K_0$ such that if $\Omega\subset\hat{\mathbb{C}}$ is a simply connected hyperbolic domain, then there exists aconformally natural $K_0$-quasiconformal map from $\Omega$ to the boundary${\rm Dome}(\Omega)$ of the convex hull of its complement which extends to theidentity on $\partial\Omega$. Explicit upper and lower bounds on $K_0$ wereobtained by Epstein, Marden, Markovic and Bishop. We improve on these bounds,by showing that one may choose $K_0\le 7.1695$.

An improved bound for Sullivan's convex hull theorem