Line segments in the isotropic planar STIT tessellation (Q2837748)

The planar STIT (stable under iteration) tessellation, first introduced by \textit{W. Nagel} and \textit{V. Weiss} [Adv. Appl. Probab. 35, No. 1, 123--138 (2003; Zbl 1023.60015); Adv. Appl. Probab. 37, No. 4, 859--883 (2005; Zbl 1098.60012)] and studied by them and others.NEWLINENEWLINEIn this paper, the author presents a characterization of the structure of the internal vertices of STIT tessellations, the so-called \(I\)-segments. An \(I\)-segment in an STIT tessellation is a convex union of tessellation edges that is not contained in a longer convex union of edges. A convex union is a union of collinear edges (which join together to make a line segment). The new characterization presented in the paper enables the computation of the probability distributions of the numbers of various types of edges and sides within a typical \(I\)-segment and the number of vertices internal to a typical cell side. As with all stationary structures, a typical geometric element is an element representative of its class. A typical element cannot be realized exactly but the sampling of one element (in an equally likely way) from those of that class wholly contained in a large ball of radius \(r\) and \(r\to \infty\). The distribution of a typical element, for example, the length distribution of a typical edge, can be determined by either ergodic theory or by a formula from the theory of Palm measures. The results of the paper are presented in the isotropic context, but much of the material remains valid for the non-isotropic STIT model.