Beschreibung:
<jats:title>Abstract</jats:title><jats:p>For a locally finite set in <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\mathbb {R}}}}^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>R</mml:mi>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula>, the order-<jats:italic>k</jats:italic> 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 <jats:italic>k</jats:italic>. As an example, a stationary Poisson point process in <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\mathbb {R}}}}^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mrow>
<mml:mi>R</mml:mi>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:math></jats:alternatives></jats:inline-formula> 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. <jats:bold>6</jats:bold>, 85–127 (1970)).</jats:p>