Abstract
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 Vaisala, 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 Lbilipschitz mappings are Apollonian bilipschitz with uniformly bounded constant.
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