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Comptes Rendus Mathématique
Volume 334, n° 4
pages 325-330 (2002)
Doi : S1631-073X(02)02261-6
Received : 3 January 2002 ;  accepted : 7 January 2002
La loi du plus petit disque contenant la cellule typique de Poisson-Voronoi
The law of the smallest disk containing the typical Poisson-Voronoi cell
 

Pierre Calka
Université Claude Bernard, Lyon 1, LaPCS, Bât. B, Domaine de Gerland, 50, avenue Tony-Garnier, 69366 Lyon cedex 07, France 

Note présentée par Jean-Pierre Kahane

Résumé

Soit R m (respectivement R M ) le rayon du plus grand (respectivement plus petit) disque centré à l'origine et inclus dans (respectivement contenant) la cellule typique de la mosaı̈que de Poisson-Voronoi deux-dimensionnelle. Dans ce travail, nous obtenons la loi conjointe de R m et R M . Pour cela nous faisons appel à des techniques classiques de recouvrement du cercle dûes à Stevens, Siegel et Holst ainsi qu'à une conjecture de Siegel que nous démontrons. Le calcul des probabilités conditionnelles P{RM⩾r+sRm=r} permet de préciser le caractère circulaire des cellules typiques de Poisson-Voronoi admettant un « grand » disque inscrit. Pour citer cet article : P. Calka, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 325-330.

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

Denote by R m (respectively R M ) the radius of the largest (respectively smallest) disk centered at the origin and included in (respectively containing) the typical cell of the two-dimensional Poisson-Voronoi tessellation. In this article, we obtain the joint distribution of R m and R M . This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The computation of the conditional probabilities P{RM⩾r+sRm=r} reveals the circular property of the Poisson-Voronoi typical cells having a “large” in-disk. To cite this article: P. Calka, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 325-330.

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


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