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The Apollonian Metric: Quasi-Isotropy and Seittenranta’s Metric


In this paper we consider domains in which the Apollonian metric is quasi-isotropic. We calculate optimal quasi-isotropy constants for exterior ball domains and several simple domains and show that in many cases the Apollonian metric can be globally approximated with Seittenranta’s metric using the optimal quasi-isotropy constant.

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  1. D. Barbilian, Einordnung von Lobatschewsky’s Maβbestimmung in gewisse allgemeine Metrik der Jordanschen Bereiche, Casopsis Mathematiky a Fysiky 64 (1934–35), 182–183.

    Google Scholar 

  2. A. F. Beardon, Geometry of Discrete Groups, Graduate text in mathematics 91, Springer-Verlag, New York, 1995.

    Google Scholar 

  3. A. F. Beardon, The Apollonian metric of a domain in ℝn, in: P. Duren, J. Heinonen, B. Osgood and B. Palka (eds.), Quasiconformal mappings and analysis, Springer-Verlag, New York, 1998, 91–108.

    Chapter  Google Scholar 

  4. L. M. Blumenthal, Distance Geometry. A Study of the Development of Abstract Metrics. With an introduction by Karl Menger, Univ. of Missouri Studies. Vol. 13 No. 2, Columbia, Univ. of Missouri, 1938.

    Google Scholar 

  5. W.-G. Boskoff, Hyperbolic Geometry and Barbilian Spaces, Istituto per la Ricerca di Base, Series of Monographs in Advanced Mathematics, Hardronic Press, Inc, Palm Harbor, FL, 1996.

    Google Scholar 

  6. A. F. Beardon, The Apollonian metric and quasiconformal mappings, in: I. Kra and B. Maskit (eds.), In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math. 256, Amer. Math. Soc., Providence, RI, 2000, 143–163.

    Google Scholar 

  7. F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), 50–74.

    MathSciNet  Article  MATH  Google Scholar 

  8. P. A. Hästö, The Apollonian metric: geometry of comparison and bilipschitz domains, Ph.D. Thesis (part), University of Helsinki, 2003;∼hasto/pp/.

  9. A. F. Beardon, The Apollonian metric: unifomity and quasiconvexity, Ann. Acad. Sci. Fenn. Math. 28 (2003), 385–414.

    MathSciNet  Google Scholar 

  10. A. F. Beardon, The Apollonian metric: limits of the comparison and bilipschitz properties, Abstr. Appl. Anal. 2003 (2003) 20, 1141–1158.

    Google Scholar 

  11. Z. Ibragimov, The Apollonian metric, sets of constant width and Möbius modulus of ring domains, Ph.D. Thesis, University of Michigan, Ann Arbor, 2002.

    Google Scholar 

  12. A. F. Beardon, On the Apollonian metric of domains in \(\overline{R n}\), Complex Variables 48 (2003) 10, 837–855.

    Article  Google Scholar 

  13. A. F. Beardon, Conformality of the Apollonian metric, Comput. Methods Funct. Theory 3 (2003) 2, 397–411.

    MathSciNet  Google Scholar 

  14. P. J. Kelly, Barbilian geometry and the Poincaré model, Amer. Math. Monthly 61 (1954), 311–319.

    MathSciNet  Article  MATH  Google Scholar 

  15. A. G. Rhodes, An upper bound for the hyperbolic metric of a convex domain, Bull. London Math. Soc. 29 (1997), 592–594.

    MathSciNet  Article  MATH  Google Scholar 

  16. P. Seittenranta, Möbius-invariant metrics, Math. Proc. Cambridge Philos. Soc. 125 (1999), 511–533.

    MathSciNet  Article  MATH  Google Scholar 

  17. D. A. Trotsenko, Properties of regions with a nonsmooth boundary, Sibirsk. Mat. Zh. 22 4 (1981), 221–224.

    MathSciNet  MATH  Google Scholar 

  18. M. Vuorinen, Conformal invariants and quasiregular mappings, J. Anal. Math. 45 (1985), 69–115.

    MathSciNet  Article  MATH  Google Scholar 

  19. A. F. Beardon, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics 1319, Springer-Verlag, Berlin-Heidelberg-New York, 1988.

    Google Scholar 

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Correspondence to Peter A. Hästö.

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The author was financially supported by the Finnish Academy of Science and Letters and the Academy of Finland through the Graduate School of Analysis and Logic.

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Hästö, P.A. The Apollonian Metric: Quasi-Isotropy and Seittenranta’s Metric. Comput. Methods Funct. Theory 4, 249–273 (2005).

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Key Words

  • Barbilian metric
  • quasi-isotropy
  • exterior ball domains

2000 MSC

  • 30F45
  • 30C65