The zero cell of a parametric class of random hyperplane tessellations
depending on a distance exponent and an intensity parameter is investigated, as
the space dimension tends to infinity. The model includes the zero cell of
stationary and isotropic Poisson hyperplane tessellations as well as the
typical cell of a stationary Poisson Voronoi tessellation as special cases. It
is shown that asymptotically in the space dimension, with overwhelming
probability these cells satisfy the hyperplane conjecture, if the distance
exponent and the intensity parameter are suitably chosen dimension-dependent
functions. Also the high dimensional limits of the mean number of faces are
explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In
the background are new identities linking the f-vector of the zero cell to
certain dual intrinsic volumes
