La loi du plus petit disque contenant la cellule typique de Poisson–Voronoi

Calka, Pierre

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

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 $\mathrm{P}\{{\mathrm{R}}_M⩾\mathrm{r}+\mathrm{s}\mid {\mathrm{R}}_m=\mathrm{r}\}$ permet de préciser le caractère circulaire des cellules typiques de Poisson–Voronoi admettant un « grand » disque inscrit.

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 $\mathrm{P}\{{\mathrm{R}}_M⩾\mathrm{r}+\mathrm{s}\mid {\mathrm{R}}_m=\mathrm{r}\}$ reveals the circular property of the Poisson–Voronoi typical cells having a “large” in-disk.

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