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Licensed Unlicensed Requires Authentication Published by De Gruyter June 9, 2010

The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets

P. A. Hästö

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 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.

Received: 2004-06-23
Revised: 2005-09-20
Published Online: 2010-06-09
Published in Print: 2006-December

© Heldermann Verlag

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