The Delaunay tessellation in hyperbolic space

TL;DR

This paper extends the concept of Delaunay tessellations to hyperbolic space, establishing foundational properties and duality with Voronoi diagrams, with applications to infinite and lattice-invariant sets.

Contribution

It introduces a method to construct Delaunay tessellations in hyperbolic space using convex hulls, proving key properties and addressing non-lattice-invariant cases.

Findings

Delaunay tessellation forms a polyhedral decomposition for finite and lattice-invariant sets.

Established hyperbolic versions of empty circumspheres and duality with Voronoi tessellations.

Presented examples of pathological infinite, non lattice-invariant sets.

Abstract

The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumspheres condition and geometric duality with the Voronoi tessellation are proved. Some pathological examples of infinite, non lattice-invariant sets are exhibited.