The $\beta$-Delaunay tessellation I: Description of the model and geometry of typical cells

TL;DR

This paper introduces two new classes of stationary random tessellations, the $eta$- and $eta'$-Delaunay tessellations, generalizing classical models and analyzing their typical cell properties.

Contribution

It defines the $eta$- and $eta'$-Delaunay tessellations, linking them to $eta$-polytopes and deriving key geometric characteristics.

Findings

Explicit distribution of volume-weighted typical cells

Calculation of volume moments and angle sums

Determination of cell intensities

Abstract

In this paper two new classes of stationary random simplicial tessellations, the so-called $\beta$- and $\beta'$-Delaunay tessellations, are introduced. Their construction is based on a space-time paraboloid hull process and generalizes that of the classical Poisson-Delaunay tessellation. The distribution of volume-power weighted typical cells is explicitly identified, establishing thereby a remarkable connection to the classes of $\beta$- and $\beta'$-polytopes. These representations are used to determine principal characteristics of such cells, including volume moments, expected angle sums and cell intensities.