Extremal distance, hyperbolic distance, and convex hulls over domains with smooth boundary

Given a simply connected planar domain Omega we develop estimates for boundary derivatives on partial derivative Omega and estimates for hyperbolic and extrernal distances in Omega and the hyperbolic convex hull boundary S Omega. We focus on the case when the underlying domain has smooth boundary; this allows very explicit formulas in terms of a collection of invariants which clarify behavior even in the generic case. In particular, we are able to obtain very explicit estimates using the intimate connection between the convex hull boundary and the geometry of the medial axis. As applications, we include here a refinement and alternate proof of the Thurston-Sullivan conjecture that the nearest-point retraction is 2-Lipschitz in the hyperbolic metrics and a variant of the Ahlfors distortion theorem which works as an integral along branches of the medial axis.