On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

TL;DR

This paper investigates the geometric properties of higher order Brillouin tessellations in the plane, focusing on angle distributions and monotonicity for generic, coarsely dense point sets, with implications for stochastic processes like Poisson point processes.

Contribution

It establishes the monotonic behavior of angle sequences in higher order tessellations and derives angle distributions for Poisson processes, extending classical results.

Findings

Angles in tessellations are monotonic with respect to order for generic sets.

Angle distributions in Voronoi, Delaunay, and Brillouin tessellations are order-independent.

Results apply to Poisson point processes, providing explicit angle distribution formulas.

Abstract

For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in $k$. As an example, a stationary Poisson point process in $\mathbb{R}^2$ is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-$1$ Delaunay mosaics given by Miles in 1970.