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

Abstract

Abstract For a locally finite set in $${{{\mathbb {R}}}}^2$$ 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 $${{{\mathbb {R}}}}^2$$ R 2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions 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 (Math. Biosci. 6 , 85–127 (1970)).