A Claim on Typical Voronoi Cells I am trying to prove the following claim (may be it has been proven).
Claim: Consider a set of points $\phi=\{x_1,x_2,...,x_i,...\}$ generated by a homogeneous PPP with rate $\lambda$ in the 2-D plane $\mathbb R^2$. Then we generate the Voronoi cells with the $k$ nearest points ($k$ order Voronoi cell WiKi, Demo).
Can we claim that the expected sum area of typical cells, in which a point $x_i\in \phi$, takes part is the same for any point $\{x_1,x_2,...,x_i,...\}$?
PS: Defining expectation for a point seems tricky to me. Because the points will change with each trial. Please help to formulate the expected number of cells a point $x_i$ takes part in, as a integral. I have a similar question on the number of cells but I posted it separately in order not to put too much in one question https://math.stackexchange.com/questions/612562/poisson-point-process-ppp-and-voronoi-cells
Suggestions or a references are most welcome, thanks.
 A: The 2005 paper "Statistics of Random Plane Voronoi Tessellations"
by Ken Brakke (PDF download)
computes various statistics when the points follow
a Poisson process, including edge-length distribution:
     
He quotes Gilbert for the distribution of cell areas (while Brakke computes the variance):

E. N. Gilbert, “Random subdivisions of space into crystals.” Annals of Mathematical Statistics.
  33 (1962), 958–972.

A: Notice that enumeration of points is not given a priori. The Poisson point configuration is a set of points with no order on them. It also can be viewed as a measure that puts mass 1 at each configuration point. The result is that for certain enumerations the expectation of the cell sizes will misbehave. 
Here is a very crude example. Let $x_1$ be the point that is closest to the origin among those with the cell area less than 1; let $x_2$ be the closest point to the origin among those with the cell area greater than 1; find $x_3$ and $x_4$ in the same manner among the remaining points, and so on. Clearly, the area sizes of all odd cells are less than 1, area sizes of all even cells will be greater than 1, so expectations clearly differ. You may say that this is a ridiculous way to label points, but it demonstrates the problem. 
For many other enumerations you will have identical expectations. On the second thought, I do not know any labeling that would not introduce some bias, except random labeling in a finite volume. Maybe there is a standard way in stochastic geometry to define enumeration or the expectation of the cell area without referring to enumerations at all.
I do not know much about stochastic geometry although there are several books with that title. I hope they do not overlook this issue. For point processes per se try the book by Daley and Vere-Jones. Kingman's book on Poisson process is mostly devoted to the 1-dimensional case, but is also a nice introduction.
