  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.
