We consider a stationary Poisson process X of k-flats in ℝd with intensity measure Θ and a measurable set S of k-flats depending on F1,…,Fn∈ X, x∈ℝd, and X in a specific equivariant way. If (F1,…,Fn,x) is properly sampled (in a ‘typical way’) then Θ(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 Poisson–Voronoi 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.

Distance measurements are useful tools in stochastic geometry. For a Boolean model Z in ℝd, the classical contact distribution functions allow the estimation of important geometric parameters of Z. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random sets Z which are generated as the union sets of Poisson processes X of k-flats, k ∈ {0, …, d-1}, and study distances from a fixed point or a fixed convex body to Z. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members of X and investigate the associated process of nearest points in the flats of X. In particular, we discuss to which extent the directional distribution of X is determined by this point process. Some of our results are presented for more general stationary processes of flats.