Communications in Analysis and Geometry
Volume 12 (2004)
The Apollonian Inner Metric
Pages: 927 – 947
In this paper we derive an explicit formula for the inner metric of the Apollonian metric and prove that for most domains there exists a geodesic connecting two arbitrary points. We also give a necessary and sufficient condition for the Apollonian inner metric to be bilipschitz equivalent to the quasihyperbolic metric.