A mosaic (or tessellation) is a locally finite subdivision of real space into bounded convex polytops. Random
mosaics have attracted considerable interest since the last century and have become of increasing importance
during the last twenty years due to applications in different fields such as biology, material science, medicine, or
telecommunication. Among the large variety of models studied today Poisson hyperplane tessellations and Poisson
Voronoi mosaics play a crucial role and are the focal point of this project.
One of the main problems in this field is to describe the shape of the cells - either the shape of the typical cell or
the shape of the cell containing the origin.
In the early 1940s, David Kendall first formulated his famous conjecture, that the conditional law for the shape of
extremal cells of a mosaic, given the area of this cell is large, converges weakly to the degenerate law concentrated
at the circular shape. Only in the last years these questions were treated successfully by Kovalenko, and Hug,
Schneider, and Reitzner. In all these cases the size of the cells are measured by volumes, intrinsic volumes, or
diameter, i.e. by metric functionals.
In this project the interesting case of cells with a large number of vertices or facets should be investigated: Does
the conditional law for the shape of "extremal" cells of a mosaic, meaning that the number of faces (vertices) of
this cell is large, converge weakly to a degenerate law? Thus, in a first step we will study mean values of
combinatorial characteristics of the cero cell and the typical cell of a stationary Poisson hyperplane tessellation and
stationary Poisson Voronoi tesselation. Next, distributional properties of the latter will be investigated, and in a
third step the obtained information on the combinatorial characteristics will hopefully lead to new insight in the
behaviour of extremal cells with a large number of facets, vertices, or faces of other dimension.