What's the Distance Distribution for Points within the Poisson-Voronoi Tessellation I would like to ask a question about the distance distribution (from the origin/core to any location within its Poisson-Voronoi Tessellation).
We read some about Tessellations and find there is description of the contact density function, the intersection length distribution, etc.
But what is the distance distribution for any point(location) within the voronoi tessellation to the origin?
Could you tell where to find it or give me some tips?
Thanks in advance. Comments and discussions are welcome if I didn't state clearly.
 A: Here is how I interpret the intended meaning of this question:
Suppose we have a two-dimensional (people usually mean 2D when they talk about Voronoi tessellations, although they can exist in any number of dimensions) Poisson Voronoi tessellation with a density $\rho$ of seed points. To each seed point is associated a cell. If we select a point uniformly and at random from within a given seed point's cell, what is the probability density function (pdf) for the distance between the random point and that cell's seed point?
Clearly, each cell has its own size and shape, and therefore its own unique distance pdf. What we really want is an average, generic pdf, averaged over all such cells. So we instead imagine picking a random point $p$ uniformly from anywhere in the whole tessellation (which we'll take to be arbitrarily large to avoid edge effects), and then determine the distance to the seed of whatever cell $p$ happens to land in.
However, by the definition of a Voronoi tessellation this is entirely equivalent to simply finding the pdf of the distance between $p$ and the nearest seed point... also known as the nearest-neighbor problem.
Without loss of generality, take the point $p$ to be the origin. We imagine surrounding the origin with a finite circle of radius $R$ (which we'll later take to infinity) containing $N = \pi R^2 \rho$ seed points, for a density of $\rho$ within the circle. The cumulative distribution function (cdf) of the distance $r$ between each independently-selected seed point and the origin is given by
$$
F(r) = r^2 / R^2 \quad \text{for $0<r<R$.}
$$
Then the associated pdf is
$$
f(r) = F'(r) = 2r/R^2 \quad \text{for $0<r<R$.}
$$
What we want is the pdf for the distance to the origin of the nearest of these $N$ points. This is the $k = 1$ order statistic of these $N$ samples, and its pdf is given by (see link):
$$
f_1(r) = N {\left(1 - \frac{r^2}{R^2}\right)}^{N-1} \frac{2r}{R^2}\, .
$$
If we use the fact that $R^2 = N / \pi\rho$, this becomes:
$$
f_1(r) = 2\pi\,\rho\, r {\left(1 - \frac{\pi\rho r^2}{N}\right)}^{N-1}\, .
$$
Finally, we take the limit as $N\rightarrow\infty$, since this is an "arbitrarily-large" tesselation, to arrive at the answer to the original question:
$$
f_1(r) \rightarrow 2\pi\rho\,  r\, \mathrm{e}^{- \pi \rho r^2}\, .
$$
All this assuming I've interpreted the question correctly. By the way, if you have access to a university or other technical library, the standard reference for all things Voronoi is this book by Okabe, et al.
