Second moments related to Poisson hyperplane tessellations

It is well known that the vertex number of the typical k-face of a stationary random hyperplane tessellation in Rd has, under some mild conditions, an expectation equal to 2k, independent of the underlying distribution. The variance of this vertex number can vary widely. Under Poisson assumptions, we give sharp bounds for this variance, showing, in particular, that its maximum is attained if the tessellation is isotropic (that is, its distribution is rotation invariant) with respect to a suitable scalar product on Rd. The employed representation of the second moment of the vertex number is a special case of formulas providing the covariance matrix of the random vector (ℓ0,…,ℓk), where ℓr is the total r-face content of the typical k-face of a stationary Poisson hyperplane mosaic. For k=d in the isotropic case, such formulas were first obtained by Miles in 1961. We give a more elementary proof and extend the formulas to general orientation distributions and to k-dimensional faces.