On the relationship between the 0-cell and the typical cell of a stationary random tessellation

摘要

The paper contains variations on the subject, “The 0-cell in a stationary random tessellation is in a sense larger than the typical cell”. It is well-known that the mean volume of the 0-cell in a stationary random tessellation in the d-dimensional Euclidean space (d=1,2,…) is not smaller than the mean volume of the typical cell. It can be shown that a corresponding statement is true for all moments. By a suitable interpretation, it is possible to say that almost surely the volume of the typical cell is not greater than the volume of the 0-cell. In the case of Poisson hyperplane tessellation, it is even true that the typical cell can be imbedded as a subset in the 0-cell.