Assuming general position, I want to prove that for a Voronoi diagram of $n$ points, the average number of vertices of a cell is arbitrarily close to $6$ as $n → ∞$.
I am not really sure how I should do it. One possible way is to do it by probabilistic method.
Any help will be appericiated.
It's not true in general. If the points are arranged in a square grid, the average number of vertices per cell will be 4.
However, for a random set of points, it's true. You can start by proving that the degree of a vertex is almost surely 3. After that you can use the Euler formula for a planar graph.