The Hilbert metric is a distance function defined for points lying within the interior of a convex body. It arises in the analysis and processing of convex bodies, machine learning, and quantum information theory. In this paper, we show how to adapt the Euclidean Delaunay triangulation to the Hilbert geometry defined by a convex polygon in the plane. We analyze the geometric properties of the Hilbert Delaunay triangulation, which has some notable differences with respect to the Euclidean case, including the fact that the triangulation does not necessarily cover the convex hull of the point set. We also introduce the notion of a Hilbert ball at infinity, which is a Hilbert metric ball centered on the boundary of the convex polygon. We present a simple randomized incremental algorithm that computes the Hilbert Delaunay triangulation for a set of $n$ points in the Hilbert geometry defined by a convex $m$-gon. The algorithm runs in $O(n (\log n + \log^3 m))$ expected time. In addition we introduce the notion of the Hilbert hull of a set of points, which we define to be the region covered by their Hilbert Delaunay triangulation. We present an algorithm for computing the Hilbert hull in time $O(n h \log^2 m)$, where $h$ is the number of points on the hull's boundary.

翻译：希尔伯特度量是为位于凸体内部的点定义的距离函数，它出现在凸体分析与处理、机器学习以及量子信息论中。本文展示了如何将欧几里得德劳内三角剖分适应于由平面凸多边形定义的希尔伯特几何。我们分析了希尔伯特德劳内三角剖分的几何性质，其与欧几里得情形存在一些显著差异，包括该三角剖分不一定覆盖点集的凸包。我们还引入了无穷远希尔伯特球的概念，即中心位于凸多边形边界上的希尔伯特度量球。我们提出了一种简单的随机增量算法，用于在由凸$m$边形定义的希尔伯特几何中计算$n$个点的希尔伯特德劳内三角剖分。该算法的期望运行时间为$O(n (\log n + \log^3 m))$。此外，我们引入了点集希尔伯特凸包的概念，将其定义为希尔伯特德劳内三角剖分所覆盖的区域。我们提出了在$O(n h \log^2 m)$时间内计算希尔伯特凸包的算法，其中$h$为凸包边界上的点数。