The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions

Gusakova, Anna; Kabluchko, Zakhar; Thäle, Christoph

Abstract

The β-Delaunay tessellation in Rd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a β-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in Rd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. 2010 Mathematics Subject Classification. 52A22, 52A40, 60D05, 60F05, 60F10.