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Gromov hyperbolicity of the Hilbert distance

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Abstract

In this paper, we investigate Bonk–Schramm pseudodistances on bounded domains in \({\mathbb {R}}^n\) with \({\mathcal {C}}^2\)-smooth boundary. We use them to propose an alternative proof of the Gromov hyperbolicity of strongly convex domains in \({\mathbb {R}}^n\), emphasizing the similarities between the Kobayashi distance and the Hilbert distance and unifying the results in Balogh and Bonk (Comment Math Helv 75:504–533, 2000), Karlsson and Noskov (Enseign Math II 48:73–89, 2002). This approach is inspired by the works of Bonk and Schramm (Geom Funct Anal 10:266–306, 2000), Balogh and Bonk (Comment Math Helv 75:504–533, 2000) and Blanc-Centi (Math Z 263:481–498, 2009).

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References

  1. Balogh, Z.M., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75(3), 504–533 (2000)

    Article  MathSciNet  Google Scholar 

  2. Blanc-Centi, I.: On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case. Math. Z. 263, 481–498 (2009)

    Article  MathSciNet  Google Scholar 

  3. Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal 10, 266–306 (2000)

    Article  MathSciNet  Google Scholar 

  4. Brooks, J.N., Strantzen, J.B.: Blaschke’s rolling theorem in \({\mathbb{R}}\). Mem. Amer. Math. Soc. 405,(1989)

  5. Colbois, B., Verovic, P.: Hilbert geometry for strictly convex domains. Geometriae Dedicata 105, 29–42 (2004)

    Article  MathSciNet  Google Scholar 

  6. Delgado, J.A.: Blaschke’s theorem for convex hypersurfaces. J. Differ. Goem. 14, 489–496 (1979)

    MathSciNet  MATH  Google Scholar 

  7. Ghys, E., de la Harpe, P.: Sur les groupes hyperboliques d’apres Mikhael Gromov Progr. Math., vol. 83. Birkhauser, Boston (1990)

    Book  Google Scholar 

  8. Gromov, M., Lafontaine, J., Pansu, P.: Structures métriques pour les variétés riemanniennes. Cedic/F. Nathan, Paris (1981)

    Google Scholar 

  9. Karlsson, A., Noskov, G.: The Hilbert metric and Gromov hyperbolicity. Enseign. Math. II(48), 73–89 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Koutroufiotis, D.: On Blaschke’s rolling theorems. Arch. Math. 23, 655–660 (1972)

    Article  MathSciNet  Google Scholar 

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  1. University Monastir Institut préparatoire aux etudes, d’ingénieur de Monastir, 5000, Monastir, Tunisia

    Fathi Haggui & Houcine Guermazi

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  1. Fathi Haggui
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  2. Houcine Guermazi
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Correspondence to Fathi Haggui.

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Haggui, F., Guermazi, H. Gromov hyperbolicity of the Hilbert distance. Ann Glob Anal Geom 61, 235–251 (2022). https://doi.org/10.1007/s10455-021-09809-x

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  • DOI: https://doi.org/10.1007/s10455-021-09809-x

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