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]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 325-330.

Denote by Rm (respectively RM) 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 Rm and RM. This result is derived from the covering properties of the circle due to Stevens, Siegel and Holst. The computation of the conditional probabilities P{R M r+sR m =r} reveals the circular property of the Poisson–Voronoi typical cells having a “large” in-disk.

Soit Rm (respectivement RM) 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 Rm et RM. 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{R M r+sR m =r} permet de préciser le caractère circulaire des cellules typiques de Poisson–Voronoi admettant un « grand » disque inscrit.

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DOI: 10.1016/S1631-073X(02)02261-6
Calka, Pierre 1

1 Université Claude Bernard, Lyon 1, LaPCS, Bât. B, Domaine de Gerland, 50, avenue Tony-Garnier, 69366 Lyon cedex 07, France
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Calka, Pierre. La loi du plus petit disque contenant la cellule typique de Poisson–Voronoi. Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 325-330. doi : 10.1016/S1631-073X(02)02261-6. http://www.numdam.org/articles/10.1016/S1631-073X(02)02261-6/

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