Abstract and Applied Analysis
Volume 2003 (2003), Issue 20, Pages 1141-1158

The Apollonian metric: limits of the comparison and bilipschitz properties

Peter A. Hästö

Department of Mathematics, University of Michigan, East Hall, Ann Arbor 48109-1109, MI, USA

Received 12 July 2003

Copyright © 2003 Peter A. Hästö. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Apollonian metric is a generalization of the hyperbolic metric. It is defined in arbitrary domains in n. In this paper, we derive optimal comparison results between this metric and the jG metric in a large class of domains. These results allow us to prove that Euclidean bilipschitz mappings have small Apollonian bilipschitz constants in a domain G if and only if G is a ball or half-space.