TITLE: "Average" Voronoi diagrams without probability?
QUESTION [5 upvotes]: A plane Poisson process with uniform intensity scatters "sites" about the plane.  If I'm not mistaken, in a sense the "average" Voronoi diagram of that set of sites is a honeycomb.  I know it's been proved that the average number of edges of the cells is $6$, and I've read (but not in anything published very recently) that the probabilities that the number of edges is equal to $n$ for $n\in\{3,4,5,\ldots\}$ is known only numerically.
My question is whether there is any sense in which the average shape could be said to be a honeycomb or the number of edges in an average Voronoi diagram can be shown to be $6$ without any sort of probability distribution on the set of sites?
This is a vague hunch which I suspect was inspired in part by the time I read in Seymour Geisser's book on predictive inference a way of deriving Student's small-sample confidence intervals without using probability.  Although probability was not mentioned, the squaring function as an objective function was relied on.

REPLY [3 votes]: There is a theorem [in a short three author manuscript  to be posted soon on archive] that proves:
Given any decomposition of the plane into topological cells satisfying a rather weak geometric condition, the average number of sides in a scaled up region of reasonable shape is defined in the limit and equal to a number at most six.
The  geometric condition is: each cell has a diameter bounded from above and each cell has an area bounded from below. 
Each condition is necessary [examples of Chris Bishop at Stony Brook math]. 
Also in 3D there are convex cell decompositions with arbitrarily large average face numbers  with all of the interior bodies congruent.[example of Mike Wigler Cold Spring Harbor Labs] third author
                                               Dennis Sullivan