A Voronoi tessellation of \(\mathbb{R}^2\) is a partitioning of the plane into cells constructed around point particles in such a way that each point of the space is in the cell of the particle to which it is closest. When the point particle corresponds to a homogeneous Poisson point process, the resulting partition is said to be a Poisson-Voronoi tessellation.
A different way of partitioning the plane into cells is by means of intersecting straight lines. When these correspond to a homogeneous Poisson line process, the partition is said to be a Poisson line tessellation.
The Poisson-Voronoi tessellation and the Poisson line tessellation are both statistically invariant under translations and rotations in the plane, and, for both, the cells are convex polygons. The cell that contains the origin is generally called the zero-cell, or Crofton cell in the Poisson line tessellation. The cell property most studied is the sidedness probability \(p_n\), that is, the probability for the cell to have \(n\) sides.
Let \(a\) be a line in the plane and \(\mathbb{R}_a\) be the projection of the origin onto that line, so that \(a\) is uniquely determined by \(\mathbb{R}_a\). In this paper the projection vectors are distributed with a density \(\rho(R) = cst\times \mathbb{R}^{\alpha-2}\), with \(\alpha\geq 1\), so that a family of tessellations, depending on \(\alpha\) is obtained. The authors study the asymptotic large-\(n\) expansion of the sidedness probability \(p_n(\alpha)\) of the zero-cell. Special cases are the typical Poisson-Voronoi cell \((\alpha = 2)\), and the Crofton cell \((\alpha = 1)\). In the large-\(n\) limit the cell is shown to become circular. The circle is centered at the origin when \(\alpha>1\), but gets delocalized for the Crofton cell. The paper ends with a few interesting remarks concerning possible further works.