Spatial STIT Tessellations -- Distributional Results for I-Segments

TL;DR

This paper derives an explicit formula for the distribution of vertices in I-segments of 3D STIT tessellations, enhancing understanding of their spatial structure and construction process.

Contribution

It provides the first explicit distributional formula for vertices in I-segments of 3D STIT tessellations, along with related distributional identities.

Findings

Explicit distribution formula for vertices in I-segments

Distributional identities for typical and length-weighted I-segments

Insights into the spatio-temporal construction process of STIT tessellations

Abstract

Three-dimensional random tessellations that are stable under iteration (STIT tessellations) are considered. They arise as a result of subsequent cell division, which implies that their cells are not face-to-face. The edges of the cell-dividing polygons are the so-called I-segments of the tessellation. The main result is an explicit formula for the distribution of the number of vertices in the relative interior of the typical I-segment. On the way of its proof other distributional identities for the typical as well as for the length-weighted typical I-segment are obtained. They provide new insight into the spatio-temporal construction process.