Zero Cell and Typical Cell

With this chapter, we begin to investigate the random polytopes which are associated with a stationary Poisson hyperplane tessellation, as ‘average’ cells or faces, possibly with weights. The zero cell of the tessellation is the almost surely unique cell containing the origin. Next, we consider the ‘typical cell’. The intuitive idea is to choose a cell of the tessellation at random, with equal chances for each cell to be selected, and to translate it so that its ‘center’ coincides with the origin. The actual definition is slightly more sophisticated, but once an exact definition is given, it turns out that the typical cell has representations that come close to the heuristic interpretation. Replacing the origin by a given convex body K, the zero cell is generalized by the K-cell. This is the intersection of all closed halfspaces containing K that are bounded by hyperplanes of the process not intersecting the interior of K. The chapter investigates these different types of cells.