A Poisson point proess of constant intensity in $\mathbb R^n$ has a Voronoi diagram. It is known that when $n=2$ the average number of edges of a cell is exactly $6$. Last I heard (but that was a while ago), the probability distribution of the number of edges was not known in closed form.
It seems in some vague sense the average shape of a cell when $n=2$ is a regular hexagon.
So:
- Can the "average shape" statement be made precise and proved?
- What is the "average shape", and what is the average number of facets, when $n\ge 3$?
