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Comptes Rendus Mathématique
Volume 344, n° 8
pages 477-481 (avril 2007)
Doi : 10.1016/j.crma.2007.02.018
Received : 13 February 2007 ;  accepted : 20 February 2007
Sur le caractère borné de la cellule de Crofton des mosaïques de géodésiques dans le plan hyperbolique
Boundedness of Croftonʼs cell for the homogeneous Poisson process of geodesics in the hyperbolic plane
 

Sylvain Porret-Blanc
Unité de mathématiques pures et appliquées, École normale supérieure de Lyon, CNRS UMR 5669, 46, allée dʼItalie, 69364 Lyon cedex 07, France 

Résumé

Soit   la cellule de Crofton associée à un processus poissonnien stationnaire de géodésiques du plan hyperbolique dʼintensité . En étudiant le caractère borné de la cellule de Crofton sous la forme dʼun problème de recouvrement du cercle par des arcs aléatoires, nous trouvons que la condition nécessaire et suffisante pour que   soit bornée presque-sûrement est que  . Nous obtenons de plus dʼaprès des résultats dûs à Stevens, Siegel et Holst et déjà utilisés par Calka dans la cas euclidien, la loi du plus petit rayon du disque centré en 0 et contenant   (pour  ). Pour citer cet article : S. Porret-Blanc, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.
Abstract

Denote by   the Crofton cell of an homogeneous Poisson process of geodesics in the hyperbolic plane with intensity  . In this Note, we derive from covering properties of the circle by random arcs, that the Crofton cell   is almost-surely bounded if and only if  . Moreover, some results due to Stevens, Siegel and Holst, which have been already used by Calka in the Euclidean case, allow us to estimate the law of the radius of the smallest disc centered at origin containing  . To cite this article: S. Porret-Blanc, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.


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