  We show that the nearest point retraction is a uniform quasi-isometry from
the Thurston metric on a hyperbolic domain in the Riemann sphere to the
boundary of the convex hull of its complement. As a corollary, one obtains
explicit bounds on the quasi-isometry constant of the nearest point retraction
with respect to the Poincare metric when the domain is uniformly perfect. We
also establish Marden and Markovic's conjecture that a hyperbolic domain is
uniformly perfect if and only if the nearest point retraction is Lipschitz with
respect to the Poincare metric.
