On Voronoi diagrams in the Funk Conical Geometry

2026-02-21

AI Summary

This study addresses the construction of Voronoi diagrams under the forward and backward Funk weak metrics in Funk cone geometry. By leveraging geometric transformations, reductions to Apollonius and abstract Voronoi diagrams, and analysis of conic sections, the authors demonstrate that bisectors consist of rays emanating from the cone apex and establish an equivalence between d-dimensional Funk Voronoi diagrams and (d−1)-dimensional weighted Voronoi diagrams. The work presents the first systematic and efficient algorithms: achieving O(n^{⌈(d−1)/2⌉+1}) time complexity for d-dimensional ellipsoidal cones and O(mn log n) for three-dimensional polyhedral cones with m facets. Furthermore, it fully characterizes the conditions under which three sites in a three-dimensional cone admit a circumcenter, significantly advancing the computational theory of Funk geometry.

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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.

Problem

• Voronoi diagrams
• Funk metrics
• conical geometry
• weak metrics
• computational geometry

Innovation

• Funk metric
• Voronoi diagram
• additively-weighted Voronoi
• Apollonius diagram
• abstract Voronoi diagram