Proofs of Two Conjectures by Mecke for Mixed Line-Generated Tessellations

TL;DR

This paper proves two conjectures related to Mecke's continuous-time tessellation model, establishing key properties and their equivalence to STIT tessellations under certain conditions, advancing understanding of stochastic geometric processes.

Contribution

It provides rigorous proofs for two conjectures in Mecke's line-generated tessellation model, clarifying its properties and relation to STIT tessellations.

Findings

Proof of the first conjecture confirming expected properties.

Proof of the second conjecture establishing model characteristics.

Identification of a key property of Cowan's continuous-time model.

Abstract

For a compact and convex window, Mecke described a process of tessellations which arise from cell divisions in discrete time. At each time step, one of the existing cells is selected according to an equally-likely law. Independently, a line is thrown onto the window. If the line hits the selected cell the cell is divided. If the line does not hit the selected cell nothing happens in that time step. With a geometric distribution whose parameter depends on the time, Mecke transformed his construction into a continuous-time model. He put forward two conjectures in which he assumed this continuous-time model to have certain properties with respect to their iteration. These conjectures lead to a third conjecture which states the equivalence of the construction of STIT tessellations and Mecke's construction under some homogeneity conditions. In the present paper, the first two conjectures are proven. A key tool to do that is a property of a continuous-time version of the equally-likely model classified by Cowan.