Equivalence of the Apollonian and Its Inner Metric

We show that the equivalence of the Apollonian metric and its inner metric remains unchanged by the removal of a point from the domain. For this we need to assume that the complement of the domain is not contained in a hyperplane. This improves a result of the authors wherein the same conclusion was reached under the stronger assumption that the domain contains an exterior point.

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 inℝn. In this paper, we derive optimal comparison results between this metric and thejGmetric in a large class of domains. These results allow us to prove that Euclidean bilipschitz mappings have small Apollonian bilipschitz constants in a domainGif and only ifGis a ball or half-space.