Small faces in stationary Poisson hyperplane tessellations

TL;DR

This paper investigates the shape distribution of small faces in a stationary Poisson hyperplane tessellation, showing that as the size tends to zero, the shape converges to simplices under certain directional assumptions.

Contribution

It extends previous results by establishing the limit shape distribution for small faces in hyperplane tessellations under specific directional conditions.

Findings

Limit distribution of small face shapes is concentrated on simplices.

Results generalize Gilles Bonnet's earlier work.

Provides a theoretical foundation for understanding small face geometries.

Abstract

We consider the tessellation induced by a stationary Poisson hyperplane process in $d$-dimensional Euclidean space. Under a suitable assumption on the directional distribution, and measuring the $k$-faces of the tessellation by a suitable size functional, we determine a limit distribution for the shape of the typical $k$-face, under the condition of small size and this tending to zero. The limit distribution is concentrated on simplices. This extends a result of Gilles Bonnet.