The Diameter of a Hyperbolic Disc

Recently, Matti Vuorinen asked whether the set-theoretic diameter of a hyperbolic disc of radius r in a hyperbolic plane region Ω is 2r. The answer is affirmative if Ω is simply or doubly connected. However, there are a hyperbolic discs in the triply-punctured sphere whose set-theoretic diameter is less than twice the radius. Also, for finitely connected hyperbolic plane regions all hyperbolic discs sufficiently close to the boundary have set-theoretic diameter equal to twice the radius. Precisely, if Ω is a hyperbolic plane region of finite connectivity, then there is a compact subset K of Ω such that any hyperbolic disc which is disjoint from K has diameter equal to twice the radius.