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The Hilbert metric is a distance function defined for points lying within theinterior of a convex body. It arises in the analysis and processing of convexbodies, machine learning, and quantum information theory. In this paper, weshow how to adapt the Euclidean Delaunay triangulation to the Hilbert geometrydefined by a convex polygon in the plane. We analyze the geometric propertiesof the Hilbert Delaunay triangulation, which has some notable differences withrespect to the Euclidean case, including the fact that the triangulation doesnot necessarily cover the convex hull of the point set. We also introduce thenotion of a Hilbert ball at infinity, which is a Hilbert metric ball centeredon the boundary of the convex polygon. We present a simple randomizedincremental algorithm that computes the Hilbert Delaunay triangulation for aset of $n$ points in the Hilbert geometry defined by a convex $m$-gon. Thealgorithm runs in $O(n (\log n + \log^3 m))$ expected time. In addition weintroduce the notion of the Hilbert hull of a set of points, which we define tobe the region covered by their Hilbert Delaunay triangulation. We present analgorithm 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.

Delaunay Triangulations in the Hilbert Metric