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

TL;DR

This paper studies the geometric and probabilistic properties of $eta$-Delaunay tessellations in high dimensions, revealing shape approximations and deriving limit theorems for cell volumes as dimension grows.

Contribution

It extends the understanding of $eta$-Delaunay tessellations by analyzing shape behavior conditioned on volume and establishing high-dimensional limit theorems.

Findings

Weighted typical cells resemble regular simplices for large volume

Asymptotic volume behavior follows central limit and large deviation principles

High-dimensional limit theorems are established for cell volumes

Abstract

The $\beta$-Delaunay tessellation in $\mathbb{R}^{d-1}$ 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 $\beta$-Delaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex in $\mathbb{R}^{d-1}$. 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 $\beta$-Delaunay tessellation is analysed, as $d\to\infty$. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived.