On the size-distribution of Poisson Voronoi cells

Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices. Although this particular space tessellation is intensively studied by mathematicians, in two- and three dimensional spaces there is no exact result known for the size-distribution of Voronoi cells. Motivated by the simple form of the distribution function in the one-dimensional case, a simple and compact analytical formula is proposed for approximating the Voronoi cell's size distribution function in the practically important two- and three dimensional cases as well. Denoting the dimensionality of the space by d (d=1,2,3) the $f(y)=Const*y^{(3d-1)/2}exp(-(3d+1)y/2)$ compact form is suggested for the normalized cell-size distribution function. By using large-scale computer simulations the validity of the proposed distribution function is studied and critically discussed.