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