Large planar Poisson–Voronoi cells containing a given convex body

Calka, Pierre; Demichel, Yann; Enriquez, Nathanaël

Annales Henri Lebesgue, Tome 4 (2021), pp. 711-757.

Soit $K$ un corps convexe de ${ℝ}^2$. Nous considérons la mosaïque de Voronoï engendrée par un processus ponctuel de Poisson homogène d’intensité $\lambda$ conditionné par l’existence d’une cellule $K_{\lambda }$ contenant $K$. Quand $\lambda \to \infty$, la cellule $K_{\lambda }$ converge en décroissant vers $K$ et nous donnons les estimées asymptotiques précises des espérances de la différence d’aire, de la différence de périmètre et du nombre de sommets. Comme dans les articles fondateurs de Rényi et Sulanke sur les enveloppes convexes aléatoires, la régularité de $K$ a une importance cruciale et nous traitons séparément le cas lisse et le cas polygonal. Les méthodes sont basées sur des estimées fines de l’aire de la fleur de Voronoï et de la fonction de support de $K_{\lambda }$ et sur une relation de type Efron. Enfin, nous montrons l’existence de variances limites dans le cas lisse pour la différence d’aire et le nombre de sommets ainsi que des résultats analogues d’espérances asymptotiques pour le modèle de la cellule de Crofton.

Let $K$ be a convex body in ${ℝ}^2$. We consider the Voronoi tessellation generated by a homogeneous Poisson point process of intensity $\lambda$ conditional on the existence of a cell $K_{\lambda }$ which contains $K$. When $\lambda \to \infty$, this cell $K_{\lambda }$ converges from above to $K$ and we provide the precise asymptotics of the expectation of its defect area, defect perimeter and number of vertices. As in Rényi and Sulanke’s seminal papers on random convex hulls, the regularity of $K$ has crucial importance and we deal with both the smooth and polygonal cases. Techniques are based on accurate estimates of the area of the Voronoi flower and of the support function of $K_{\lambda }$ as well as on an Efron-type relation. Finally, we show the existence of limiting variances in the smooth case for the defect area and the number of vertices as well as analogous expectation asymptotics for the so-called Crofton cell.

Classification : 52A22, 60D05, 52A23, 60G55
Mots clés : Poisson–Voronoi tessellation, Voronoi flower, Support function, Steiner point, Efron identity

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