Auxiliary Functionals and Bodies

We assume in this chapter that a nondegenerate stationary Poisson hyperplane process is given, which has locally finite intensity measure and hence an intensity and a spherical directional distribution. The properties of the hyperplane process and its induced tessellation will in an essential way depend on the spherical directional distribution. In some cases, this dependence can be analysed via the introduction of some auxiliary convex bodies. For example, by Minkowski’s theorem, the directional distribution is the surface area measure of a convex body, and the expected number of hyperplanes of the tessellation hitting a given convex body can be expressed as a mixed volume of this body and the auxiliary body. From this fact, further conclusions can be drawn. The directional distribution can also be used to define another origin-symmetric convex body, which is the projection body of the previous one and is called the Matheron zonoid. Quantities derived from the tessellation have geometric interpretations in terms of this convex body. Known information from convex geometry yields new information about the hyperplane process and its generated tessellation.