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The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets
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P. A. Hästö
Veröffentlicht/Copyright:
9. Juni 2010
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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Artikel in diesem Heft
- and Ultrafilters on ℚ and ω
- Remarks on Best Approximations in Generalized Convex Spaces
- A Note on P-Times and Time Projections
- The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets
- On the Integral Quasicontinuity
- On Completeness of Random Transition Count for Markov Chains
- Notes on Uniqueness of Meromorphic Functions
- Strong and Weak Convergence of Implicit Iterative Process with Errors for Asymptotically Nonexpansive Mappings
- On Measurable Sierpiński-Zygmund Functions
- Extension Theorem for a Functional Equation
Schlagwörter für diesen Artikel
Apollonian metric;
Barbilian metric;
bilipschitz mappings;
quasiballs
Artikel in diesem Heft
- and Ultrafilters on ℚ and ω
- Remarks on Best Approximations in Generalized Convex Spaces
- A Note on P-Times and Time Projections
- The Apollonian Metric: The Comparison Property, Bilipschitz Mappings and Thick Sets
- On the Integral Quasicontinuity
- On Completeness of Random Transition Count for Markov Chains
- Notes on Uniqueness of Meromorphic Functions
- Strong and Weak Convergence of Implicit Iterative Process with Errors for Asymptotically Nonexpansive Mappings
- On Measurable Sierpiński-Zygmund Functions
- Extension Theorem for a Functional Equation
