Poisson-Delaunay Mosaics of Order $k$

TL;DR

This paper derives explicit formulas for the expected face counts and areas in order-$k$ Voronoi tessellations of Poisson point processes, and introduces a relaxed Morse theory approach for face counting within distance thresholds.

Contribution

It provides new explicit formulas for face counts in order-$k$ Voronoi tessellations of Poisson processes and develops a relaxed Morse theory framework for face enumeration.

Findings

Explicit formulas for expected face counts and areas per unit volume.

Development of a relaxed Morse theory for face counting within distance thresholds.

Generalization of face counting methods to include distance constraints.

Abstract

The order-$k$ Voronoi tessellation of a locally finite set $X \subseteq \mathbb{R}^n$ decomposes $\mathbb{R}^n$ into convex domains whose points have the same $k$ nearest neighbors in $X$. Assuming $X$ is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the $k$ nearest points in $X$ are within a given distance threshold.