Sur les domaines hyperboliques pour la distance intégrée de Carathéodory

Vigué, Jean-Pierre

Annales de l'Institut Fourier, Tome 46 (1996) no. 3, pp. 743-753.

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 $V$, $y\ne x$, $c_D(x,y))\ne 0$. 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, $\forall y\in V$, $y\ne x$, $c_D(x,y)\ne 0$. I also give an example of a domain $D$, $c_D^i$-hyperbolic but not $c_D$-hyperbolic.

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