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.

翻译：正向与反向Funk弱度量是凸体上的基本距离函数，它们构成了Hilbert度量和Thompson度量的基础。本文研究了多边形锥与椭圆锥中正向与反向Funk度量下的Voronoi图。我们建立了若干关键几何性质：(1) 平分线由一组从锥顶射出的射线构成；(2) $d$维正向（或反向）Funk度量中的Voronoi图等价于锥体有界截面上$(d-1)$维正向（或反向）Funk度量中的加性权重Voronoi图。基于此，我们通过转换为Apollonius图及其逆变换，提出了在$d$维椭圆锥中构建此类图的$O\big(n^{\ceil{\frac{d-1}{2}}+1}\big)$时间算法；同时通过归约到抽象Voronoi图，给出了具有$m$个面的三维多边形锥的$O(mn\log(n))$时间算法。我们还完整刻画了三维锥体中三个站点存在外接中心的条件。这是对Funk度量最早期的算法研究之一。