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We consider the three-dimensional Poisson-Voronoi tessellation and study theaverage facedness m_n of a cell known to neighbor an n-faced cell. WhereasAboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptoticexpansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazaret al., based on a very large data set, now clearly rule out Aboav's law. Inthis work we determine the numerical value of k_1 and compare the expansion tothe Monte Carlo data. The calculation of k_1 involves an auxiliary planarcellular structure composed of circular arcs, that we will call thePoisson-Moebius diagram. It is a special case of more general Moebius diagrams(or multiplicatively weighted power diagrams) and is of interest for its ownsake. We obtain exact results for the total edge length per unit area, which isa prerequisite for the coefficient k_1, and a few other quantities in thisdiagram.

Large-n conditional facedness m_n of 3D Poisson-Voronoi cells