Stochastic geometry deals with random structures such as random closed sets, random processes of flats or random tessellations. A useful method for analyzing such structures is to associate a deterministic convex set (sometimes a zonoid) with it. Thus strong results from convex geometric analysis become available. As appetizers, we give two examples:

Let Z0 denote the zero cell of a stationary Poisson hyperplane tessellation. We are interested in sharp bounds for the expected number of vertices of Z0. Such bounds are provided by the Blaschke-Santaló inequality and by the Mahler inequality for zonoids. Equality cases in these bounds characterize special direction distributions of the given hyperplane tessellation. Recently, these bounds have been improved by corresponding stability estimates, first in the geometric and then in the probabilistic framework. (joint work with Károly Böröczky)

As a second, new example, let X denote a stationary Poisson hyperplane process with fixed intensity g in \Rn. From X we pass to the intersection process X(k) of order k, which is a stationary process of (n-k)-flats in \Rn. It is well known that the intersection density g(k)(X), i.e. the intensity of X(k), is maximal if and only if X is isotropic. Here we introduce a measure for the strength of intersections from an affine-invariant point of view. The problem of determining its minimal value leads to a novel geometric inequality for mixed volumes of zonoids with isotropic generating measures. The solution of a related problem involving joint intersections from a process of lines and an independent process of hyperplanes is partly based on Keith Ball's reverse isoperimetric inequality together with the equality conditions due to Franck Barthe. (joint work with Rolf Schneider)