Birth-time distributions of weighted polytopes in STIT tessellations

The lower-dimensional maximal polytopes associated with an iteration stable (STIT) tessellation in $\RR^d$ are considered. They arise in the spatio-temporal construction process of such a tessellation as intersections of $(d-1)$-dimensional maximal polytopes. A precise description of the joint distribution of their birth-times is obtained. This in turn is used to determine the probabilities that the typical or the length-weighted typical maximal segment of the tessellation contains a fixed number of internal vertices.