The combinatorial structure of spatial STIT tessellations

TL;DR

This paper provides a detailed combinatorial analysis of 3D spatial STIT tessellations, revealing new geometric mean values and classifying edges based on endpoint types, advancing understanding of their complex structure.

Contribution

It introduces a comprehensive classification of edges and computes new geometric mean values for spatial STIT tessellations, enhancing the understanding of their combinatorial structure.

Findings

New geometric mean values for neighborhood of typical vertices

Classification of tessellation edges by endpoint types

Distributional properties of spatial STIT tessellations

Abstract

Spatially homogeneous random tessellations that are stable under iteration (nesting) in the 3-dimensional Euclidean space are considered, so-called STIT tessellations. They arise as outcome of a spatio-temporal process of subsequent cell division and consequently they are not facet-to-facet. The intent of this paper is to develop a detailed analysis of the combinatorial structure of such tessellations and to determine a number of new geometric mean values, for example for the neighborhood of the typical vertex. The heart of the results is a fine classification of tessellation edges based on the type of their endpoints or on the equality relationship with other types of line segments. In the background of the proofs are delicate distributional properties of spatial STIT tessellations.