Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 60: Issue 2-3
Hilbert Metric in the Unit Ball
Authors:
Oona Rainio University of Turku, FI-20014 Turku, Finland

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https://orcid.org/0000-0002-7775-7656
and
Matti Vuorinen University of Turku, FI-20014 Turku, Finland

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https://orcid.org/0000-0002-1734-8228
Pages:
175–191
Online Publication Date:
24 Oct 2023
Publication Date:
24 Oct 2023
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2023.01544
Keywords:
Cayley–Klein metric; Hilbert metric; hyperbolic metric; Klein-Beltrami model; Klein metric
Restricted access

The Hilbert metric between two points 𝑥, 𝑦 in a bounded convex domain 𝐺 is defined as the logarithm of the cross-ratio 𝑥, 𝑦 and the intersection points of the Euclidean line passing through the points 𝑥, 𝑦 and the boundary of the domain. Here, we study this metric in the case of the unit ball 𝔹𝑛. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal and quasiregular mappings.

  • [1]

    G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen. Conformal invariants, inequalities and quasiconformal maps. J. Wiley, 1997.

  • [2]

    A .F. Beardon. The Klein, Hilbert, and Poincaré metrics of a domain, J. Comput. Appl. Math., 105:155–162, 1999.

  • [3]

    A. F. Beardon. The Apollonian metric on domains in ℝ𝑛. In P. Duren, J. Heinonen, B. Os-good, B. Palka eds., Quasiconformal Mappings and Analysis: Articles Dedicated to Frederick W. Gehring on the Occasion of his 70th Birthday. Springer, Berlin, 1998, pp. 91–108.

    • Search Google Scholar
    • Export Citation
  • [4]

    K. Böröczky, G. Kertész, and E. Makai. The minimum area of a simple polygon with given side lengths. Period Math Hung, 39:33–49, 2000.

    • Search Google Scholar
    • Export Citation
  • [5]

    J. Chen, P. Hariri, R. Klén and M. Vuorinen. Lipschitz conditions, triangular ratio metric, and quasiconformal maps. Ann. Acad. Sci. Fenn. Math., 40:683–709, 2015.

    • Search Google Scholar
    • Export Citation
  • [6]

    R. Frigerio and M. Moraschini. On volumes of truncated tetrahedra with constrained edge lengths. Period Math Hung, 79:32–49, 2019.

  • [7]

    F. W. Gehring and B. P. Palka. Quasiconformally homogeneous domains, J. Analyse Math., 30:172–199, 1976.

  • [8]

    P. Hariri, R. Klén and M. Vuorinen. Conformally Invariant Metrics and Quasiconformal Mappings. Springer, 2020.

  • [9]

    D. Hilbert. Ueber die gerade Linie als kurzeste Verbindung zweier Punkte. Math. Ann., 46:91–96, 1895.

  • [10]

    Á. Horvath. Hyperbolic plane geometry revisited. J. Geom., 106(2):341–362, 2015.

  • [11]

    Ž. Milin Šipuš. Translation surfaces of constant curvatures in a simply isotropic space. Period Math Hung 68:160–175, (2014).

  • [12]

    O. Rainio. Intrinsic metrics in ring domains. Complex Anal Synerg, 8:article no. 3, (2022).

  • [13]

    O. Rainio. Intrinsic metrics under conformal and quasiregular mappings. Publ. Math. Debrecen, 101(1-2):189–215, 2022.

  • [14]

    O. Rainio and M. Vuorinen. Triangular ratio metric in the unit disk. Complex Var. Elliptic Equ., 67(6):1299–1325, 2022.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

Managing Editor

Gábor SÁGI (Rényi Institute of Mathematics)

Editorial Board

  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
  • Penny HAXELL (University of Waterloo)
  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
  • Ron HOLZMAN (Technion, Haifa)
  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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