On Voronoi diagrams in the Funk Conical Geometry

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.

Figures (9)

• Figure 1: (a) An elliptical cone. (b) A polygonal cone.
• Figure 2: Two points $a$ and $b$ where (a) $F_C(a,b)<0$, (b) $F_C(a,b)=0$, (c) $F_C(a,b)>0$.
• Figure 3: Computing $F_C(a,b)$: (a) scaling $b$ to $\partial C$, (b) using a ratio of distances to the boundary, (c) using the distance of $a$ to the boundary.
• Figure 4: (a) Voronoi cells in the reverse Funk metric, and (b) in the forward Funk metric.
• Figure 5: (a) Reverse Funk ball tangent to two forward Funk balls, (b) forward Funk ball tangent to two reverse Funk balls.

Theorems & Definitions (44)

• Definition 1: Metric Space
• Definition 2: Conical Partial Order
• Definition 3: Forward Funk metric
• Definition 4: Reverse Funk metric
• Proposition 5
• Lemma 6
• Lemma 7
• Corollary 8
• Lemma 9
• Corollary 10