An improved bound for Sullivan's convex hull theorem

Abstract

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 $\mathrm{Dome}(\Omega )$ 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 $K_0\le 7.1695$.

24 pages, 5 figures