Observations Inside a Window

A common approach to investigating a random set or a process of convex objects is to observe it in an observation window (often convex), to draw some conclusions, and then to increase the window. An example is the determination of a functional density for a stationary particle process by means of a limit relation. The present chapter deals with the simplest questions one can ask when a stationary Poisson hyperplane process is observed inside a given bounded set, for example the question for the number of hyperplanes meeting a convex body or the number of intersection points inside a Borel set. First and second moments of such random variables will be determined. It will be explored how the observation window and the directional distribution of the hyperplane process affect the results.