The Apollonian metric: limits of the comparison and bilipschitz properties

Abstract

The Apollonian metric is a generalization of the hyperbolic metric. It is defined in arbitrary domains in ${ℝ}^n$. In this paper, we derive optimal comparison results between this metric and the $j_G$ 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.