Faces of Poisson-Voronoi mosaics (Q644789)

For a stationary Poisson process in \(\mathbb{R}^d\) the distribution of the typical \(k\)-face of the induced Voronoi tessellation is considered. Such a \(k\)-face \(F\) is (in general) the intersection of \(d-k+1\) cells of the tessellation. Their nuclei lie on the relative interior of a \((d-k)\)-dimensional ball, whose radius is called co-radius of \(F\), while its center is the generalized nucleus of \(F\). Representations are determined for the joint distribution of the typical \(k\)-face and its co-radius, as well as for the regular conditional distribution of the typical \(k\)-face, given the co-radius. This is formulated in terms of the intersection of a Poisson process of \(k\)-dimensional closed half-spaces, whose intensity measure is explicitly obtained. From this result some interesting consequences can be derived, for instance, the probability that the typical \(k\)-face contains its generalized nucleus can be computed. Also, under the condition of large inradius, it is shown that the relative boundary of the typical \(k\)-face lies, with high probability, in a narrow spherical annulus