Large cells in Poisson–Delaunay tessellations

Abstract. It is proved that the shape of the typical cell of a Delaunay tessellation, derived from a stationary Poisson point process in d-dimensional Euclidean space, tends to the shape of a regular simplex, given that the volume of the typical cell tends to infinity. This follows from an estimate for the probability that the typical cell deviates by a given amount from regularity, given that its volume is large. As a tool for the proof, a stability result for simplices is established. 1 Introduction and main result Voronoi tessellations (also called Voronoi mosaics or Voronoi diagrams) and their du-als, Delaunay tessellations, are a thoroughly studied subject of discrete geometry. The book by Okabe, Boots, Sugihara and Chiu [11] gives an impression of the richness of the theory of these tessellations and of the variety of their applications. If the discret