Large planar Poisson–Voronoi cells containing a given convex body

Abstract

Let K be a convex body in R 2 . We consider the Voronoi tessellation generated by a homogeneous Poisson point process of intensity λ conditional on the existence of a cell K λ which contains K. When λ → ∞, this cell K λ 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

Elongated Poisson-Voronoi cells in an empty half-plane

“…where δ = 2 3 ε 0 − ε 1 − 3ε 2 > 0 for ε 1 , ε 2 > 0 small enough, we completes the proof of Lemma 4.2 using (79).…”

“…Large Voronoi cells appear in locations where there is only one nucleus inside a large domain. In [2], we consider the Poisson-Voronoi tessellation generated by the union of a Poisson point process outside a large deterministic set with an isolated point belonging to that set. We then estimate the mean and variance of the area, perimeter and number of vertices of the Poisson-Voronoi cell associated with the isolated nucleus.…”