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The Apollonian metric: the comparison property, bilipschitz mappings and thick sets

Hästö P. A.

Journal of Applied Analysis

2006

Vol. 12, nr 2

209-232

The Apollonian metric is a generalization of the hyperbolic metric to arbitrary open sets in Euclidean spaces. In this article we show that the Apollonian metric is comparable, to the jG metric in the set G if and only if its complement is unbounded and thick in the sense of Väisälä Vuorinen and Wallin [Thick sets and quasisymmetric maps, Nagoya Math. J. 135 (1994), 121-148] . These conditions are also equivalent to the following: there exists L > 1 such that all Euclidean L-bilipschitz mappings are Apollonian bilipschitz with uniformly bounded constant.