Intrinsic volumes of the maximal polytope process in higher dimensional STIT tessellations

Stationary and isotropic iteration stable random tessellations are considered, which are constructed by a random process of iterative cell division. The collection of maximal polytopes at a fixed time t within a convex window is regarded and formulas for mean values, variances and a characterization of certain covariance measures are proved. The focus is on the case d>=3, which is different from the planar one, treated separately in Schreiber and Thäle (2010) [12]. Moreover, a limit theorem for suitably rescaled intrinsic volumes is established, leading -- in sharp contrast to the situation in the plane -- to a non-Gaussian limit.