Birth-time distributions of weighted polytopes in STIT tessellations

TL;DR

This paper analyzes the birth-time distributions of lower-dimensional polytopes in STIT tessellations, providing precise joint distributions and probabilities related to their internal vertices, advancing understanding of their temporal structure.

Contribution

It offers a detailed description of the joint distribution of birth-times of polytopes in STIT tessellations, a novel contribution to stochastic geometry.

Findings

Derived the joint distribution of birth-times for polytopes

Calculated probabilities for maximal segments containing a fixed number of vertices

Enhanced understanding of the temporal evolution of STIT tessellations

Abstract

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.