We consider the Voronoi tessellation based on a stationary Poisson process N in R d . We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d − k + 1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

We consider a stationary Poisson process X of k-flats in R d with intensity measure and a measurable set S of k-flats depending on F 1 , . . . , F n ∈ X, x ∈ R d , and X in a specific equivariant way. If (F 1 , . . . , F n , x) is properly sampled (in a 'typical way') then (S) has a gamma distribution. This result generalizes and unifies earlier work by Miles (1971), Møller and Zuyev (1996), and Zuyev (1999). As a new example, we will show that the volume of the fundamental region of a typical j -face of a stationary PoissonVoronoi tessellation is conditionally gamma distributed. This is true in the area-biased and the area-debiased cases. In the first case the shape parameter is not integer valued. As another new example, we will show that the generalized integral-geometric contents of the (area-biased and area-debiased) typical j -face of a Poisson hyperplane tessellation are conditionally gamma distributed. In the isotropic case the contents boil down to the mean breadth of the face.

We consider a stationary Poisson process Φ of k-flats in R d with intensity measure Λ and a random closed set S of k-flats depending on F 1 , . . . , F n ∈ Φ, x ∈ R d , and Φ in a specific equivariant way. If (F 1 , . . . , F n , x) is properly sampled, then Λ(S) has a gamma distribution. This result is generalizing and unifying earlier work in [5], [9], and [13]. As a new example we will show that the volume of the fundamental region of a typical j-face of a stationary Poisson Voronoi-tessellation is conditionally gamma-distributed.

We consider the Voronoi tessellation based on a stationary Poisson process N in ℝd. We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d−k+1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

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