The Apollonian metric: limits of the comparison and bilipschitz properties
The Apollonian metric is a generalization of the hyperbolic metric. It is defined in arbitrary domains in . In this paper, we derive optimal comparison results between this metric and the 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 if and only if is a ball or half-space.