Sur les domaines hyperboliques pour la distance intégrée de Carathéodory

Vigué, Jean-Pierre

Annales de l'Institut Fourier, Tome 46 (1996), p. 743-753 / Harvested from Numdam

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Résumé
Abstract

Dans cet article, je montre qu’un domaine $D$ est hyperbolique pour la pseudodistance intégrée de Carathéodory $c_{D}^{i}$ (c’est-à-dire que $c_{D}^{i}$ est une distance sur $D$ ) si et seulement si la pseudodistance de Carathéodory $c_{D}$ vérifie la propriété de séparation faible suivante : tout point $x$ de $D$ possède un voisinage $V$ tel que, pour tout point $y$ de $V$ , $y \neq x$ , $c_{D} (x, y)) \neq 0$ . Je construis aussi un exemple d’un domaine $c_{D}^{i}$ -hyperbolique et non $c_{D}$ -hyperbolique.

In this paper, I prove that a domain $D$ is hyperbolic for the Carathéodory integrated pseudodistance $c_{D}^{i}$ (this means that $c_{D}^{i}$ is a distance on $D$ ) if and only if the Carathéodory pseudodistance $c_{D}$ satisfies the following weak separation condition: for every $x \in D$ , there exists a neighborhood $V$ of $x$ such that, $\forall y \in V$ , $y \neq x$ , $c_{D} (x, y) \neq 0$ . I also give an example of a domain $D$ , $c_{D}^{i}$ -hyperbolic but not $c_{D}$ -hyperbolic.

Publié le : 1996-01-01

MR 97f:32031 | Zbl 0854.32010

DOI : https://doi.org/10.5802/aif.1530

@article{AIF_1996__46_3_743_0,
     author = {Vigu\'e, Jean-Pierre},
     title = {Sur les domaines hyperboliques pour la distance int\'egr\'ee de Carath\'eodory},
     journal = {Annales de l'Institut Fourier},
     volume = {46},
     year = {1996},
     pages = {743-753},
     doi = {10.5802/aif.1530},
     mrnumber = {97f:32031},
     zbl = {0854.32010},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_1996__46_3_743_0}
}

Vigué, Jean-Pierre. Sur les domaines hyperboliques pour la distance intégrée de Carathéodory. Annales de l'Institut Fourier, Tome 46 (1996) pp. 743-753. doi : 10.5802/aif.1530. http://gdmltest.u-ga.fr/item/AIF_1996__46_3_743_0/

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