On Voronoi diagrams in the Funk Conical Geometry
Aditya Acharya, Auguste Henry Gezalyan, David M. Mount, Danesh Sivakumar

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 -dimensional forward (or reverse) Funk metrics are equivalent to additively-weighted Voronoi diagrams in the -dimensional forward (or reverse) Funk metrics on bounded cross sections of the cone. Leveraging this, we provide an time algorithm for creating these diagrams in -dimensional elliptical cones using a transformation to and from Apollonius diagrams, and an time algorithm for…
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Taxonomy
TopicsPoint processes and geometric inequalities · Analytic and geometric function theory · Geometric and Algebraic Topology
