  In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert
geometries from the Finslerian viewpoint. They showed that a smooth Finsler
metric in a convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in
that domain if and only if it is complete, if its geodesics are straight lines
and if its flag curvature is equal to -1. The goal of this chapter is to
explain these notions in details, to illustrate the relation between Hilbert
geometry, Finsler geometry and the calculus of variations, and to prove the
Funk-Berwald characterization Theorem.
