  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.
