Expected Sizes of Poisson-Delaunay Mosaics and Their Discrete Morse Functions

TL;DR

This paper derives exact formulas for the expected sizes and critical features of Poisson-Delaunay mosaics in up to four dimensions using a generalized discrete Morse function.

Contribution

It introduces a new probabilistic analysis of Delaunay mosaics based on a generalized discrete Morse function for Poisson point processes.

Findings

Exact expected counts of simplices in dimensions up to 4

Formulas for expected numbers of critical simplices and non-singular intervals

Connections established with other probabilistic models

Abstract

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from an n-dimensional Poisson point process, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we get the expected numbers of simplices in the Poisson-Delaunay mosaic in dimensions up to 4.