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.