On Voronoi diagrams in the Funk Conical Geometry

TL;DR

This paper explores Voronoi diagrams under Funk metrics in convex cones, establishing geometric properties, algorithms for their construction, and characterizing circumcenters, advancing the understanding of these metrics in computational geometry.

Contribution

It introduces new geometric insights and algorithms for Voronoi diagrams in Funk conical geometries, a novel area in computational geometry.

Findings

Bisectors are rays from the cone apex.

Voronoi diagrams are equivalent to weighted diagrams on cross sections.

Efficient algorithms are provided for elliptical and polygonal cones.

Abstract

The forward and reverse Funk weak metrics are fundamental distance functions on convex bodies that serve as the building blocks for the Hilbert and Thompson metrics. In this paper we study Voronoi diagrams under the forward and reverse Funk metrics in polygonal and elliptical cones. We establish several key geometric properties: (1) bisectors consist of a set of rays emanating from the apex of the cone, and (2) Voronoi diagrams in the $d$-dimensional forward (or reverse) Funk metrics are equivalent to additively-weighted Voronoi diagrams in the $(d-1)$-dimensional forward (or reverse) Funk metrics on bounded cross sections of the cone. Leveraging this, we provide an $O\big(n^{\ceil{\frac{d-1}{2}}+1}\big)$ time algorithm for creating these diagrams in $d$-dimensional elliptical cones using a transformation to and from Apollonius diagrams, and an $O(mn\log(n))$ time algorithm for 3-dimensional polygonal cones with $m$ facets via a reduction to abstract Voronoi diagrams. We also provide a complete characterization of when three sites have a circumcenter in 3-dimensional cones. This is one of the first algorithmic studies of the Funk metrics.