Large planar Poisson–Voronoi cells containing a given convex body

Calka, Pierre; Demichel, Yann; Enriquez, Nathanaël

doi:10.5802/ahl.86

Calka, Pierre; Demichel, Yann; Enriquez, Nathanaël

Large planar Poisson–Voronoi cells containing a given convex body

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

Metadata

DOI10.5802/ahl.86

Keywords Poisson–Voronoi tessellation, Voronoi flower, Support function, Steiner point, Efron identity

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 $λ \to \infty$ , 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 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 $K_{λ}$ 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.

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