# Distance from the Nucleus to a Uniformly Random Point in the 0-cell and the Typical Cell of the Poisson-Voronoi Tessellation

## TL;DR

This paper characterizes the exact distributions of distances from the nucleus to random points in the 0-cell and typical cell of a Poisson-Voronoi tessellation, providing new integral and approximate formulas and insights into cell geometry.

## Contribution

It derives the exact and approximate distributions of these distances, linking them to contact distributions and the spherical property of large cells, advancing understanding of PV tessellation geometry.

## Key findings

- Distribution of distance to a random point in the 0-cell equals the contact distance of the Poisson process.
- Derived multi-integral expression for the distribution of the typical cell distance.
- Provided a closed-form approximate distribution with mean correction based on cell volume ratio.

## Abstract

Consider the distances $\tilde{R}_o$ and $R_o$ from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the $d$-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of $\tilde{R}_o$ and $R_o$. First, using the well-known relationship between the 0-cell and the typical cell, we show that the random variable $\tilde{R}_o$ is equivalent in distribution to the contact distance of the Poisson point process. Next, we derive a multi-integral expression for the exact distribution of $R_o$. Further, we derive a closed-form approximate expression for the distribution of $R_o$, which is the contact distribution with a mean corrected by a factor equal to the ratio of the mean volumes of the 0-cell and the typical cell. An additional outcome of our analysis is a direct proof of the well-known spherical property of the PV cells having a large inball.