Integral Geometric Tools for Stochastic Geometry

Integral geometry, as it is understood here, deals with measures on sets of geometric objects, and in particular with the determination of the total measure of various such sets having geometric significance. For example, given two convex bodies in Euclidean space, what is the total invariant measure of the set of all rigid motions which bring the first set into a position where it has nonempty intersection with the second one? Or, what is the total invariant measure of the set of all planes of a fixed dimension having nonempty intersection with a given convex body? Both questions have classical answers, known as the kinematic formula and the Crofton formula, respectively. Results of this type are useful in stochastic geometry. Basic random closed sets, the stationary and isotropic Boolean models with convex grains, are obtained by taking union sets of certain stochastic processes of convex bodies. Simple numerical parameters for the description of such Boolean models are functional densities related to the specific volume, surface area, or Euler characteristic. Kinematic formulae are indispensable tools for the investigation and estimation of such parameters. Weakening the hypotheses on Boolean models, requiring less invariance properties and admitting more general set classes, necessitates the generalization of integral geometric formulae in various directions. An introduction to the needed basic formulae and a discussion of their extensions, analogues and ramifications is the main purpose of the following. The section headings are: