# Weighted Poisson-Delaunay Mosaics

## TL;DR

This paper investigates the properties of weighted Delaunay mosaics derived from Poisson-generated Voronoi tessellations, analyzing their expected simplices and Morse intervals, and providing new insights into the topology of random geometric models.

## Contribution

It introduces a generalized discrete Morse function for weighted Delaunay mosaics and derives expected counts of simplices and Morse intervals for Poisson tessellations, offering new proofs for related topological results.

## Key findings

- Expected number of simplices as a function of radius threshold
- Expected number of Morse intervals as a function of radius
- New proof for the expected number of connected components in Boolean models

## Abstract

Slicing a Voronoi tessellation in $\mathbb{R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in $\mathbb{R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a byproduct, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in $\mathbb{R}^n$