ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2007, Vol. 42, No. 1, pp. 28–43. c  Allerton Press, Inc., 2007. STOCHASTIC GEOMETRY Length Distributions of Edges in Planar Stationary and Isotropic STIT Tessellations* J. Mecke 1** , W. Nagel 1*** , and V. Weiss 2 1 Friedrich-Schiller-Universit ¨ at Jena, Germany 2 Fachhochschule Jena, Germany Received September 2006 Abstract—Stationary and isotropic random tessellations of the euclidean plane are studied which have the characteristic property to be stable with respect to iteration (or nesting), STIT for short. Since their cells are not in a face-to-face position, three different types of linear segments appear. For all the types the distribution of the length of the typical segment is given. MSC2000 numbers: 60D05, 52A22 DOI: 10.3103/S1068362307010025 Key words: stochastic geometry, random tessellation, iteration/nesting of tessellations, stability of distributions Dedicated to the 65th Birthday of Professor R. V. Ambartzumian 1. INTRODUCTION In this paper we study the length distributions of edges of a certain class of random planar tessel- lations – the STIT tessellations. These tessellations have the characteristic property to be St able with respect to It eration (also referred to as nesting) of tessellations. The mathematical motivation for these tessellations goes back to a problem that was posed to two of the authors by R. V. Ambartzumian already in the 80-th. Also, we first learned from him the idea of the operation of iteration for tessellations. The iteration generates a new tessellation I (Y 0 , Y ) from a ’frame’ tessellation Y 0 and a sequence Y = {Y 1 ,Y 2 , ...} of independent identically distributed (i.i.d.) tessellations by subdividing the i-th cell p i of Y 0 by intersecting it with the cells of Y i , i =1, 2, ... E.g., this operation can be applied to Poisson line tessellations in the plane and it results in another tessellation. This operation of iteration can be applied repeatedly, combined with an appropriate rescaling. The problem arises whether there exists a limit tessellation when the number of repetitions goes to infinity. A further question is how such limit tessellations can be described if they exist. The existence of such tessellations was recently shown in [11], and their construction within bounded windows was described. They are STIT tessellations. In the present paper we deal with these STIT tessellations without regarding the above mentioned process of repeated rescaled iteration. An important feature of stationary STIT tessellations is that the interior of the typical cell – i.e. a random convex polygon – has the same distribution as the typical cell of a Poisson line tessellation. Hence, several results can easily be derived for stationary STIT tessellations. STIT tessellations have T-shaped nodes only, and their cells are not necessarily in a face-to-face position. Therefore, when speaking about edges or linear segments of these tessellations, it is appropriate to apply Miles’ classification which is introduced in [2], and to consider I-, J- and K-segments. ∗ The text was submitted by the authors in English. ** E-mail: mecke@minet.uni-jena.de *** E-mail: nagel@minet.uni-jena.de 28