PhD Proposal: Computational Geometry in the Hilbert Metric

The Hilbert metric generalizes the Cayley-Klein (or Beltrami-Klein) model of hyperbolic geometry to arbitrary convex polygons. It has found use in a variety of fields including graph embeddings, quantum information theory, machine learning, and convex geometry. As such there has been interest in reproducing results from classical computational geometry on the Euclidean metric to the Hilbert metric. In this document we will present our work on efficient algorithms for both Voronoi diagrams and Delaunay triangulation for point sets in the Hilbert metric. Additionally we analyze some interesting characteristics of space and bisectors in this metric. We will finish with a plan for proposed research.