  In the asymmetric setting, Hilbert's fourth problem asks to construct and
study all (non-reversible) projective Finsler metrics: Finsler metrics defined
on open, convex subsets of real projective $n$-space for which geodesics lie on
projective lines. While asymmetric norms and Funk metrics provide many examples
of essentially non-reversible projective metrics defined on proper convex
subsets of projective $n$-space, it is shown that any projective Finsler metric
defined on the whole projective $n$-space is the sum of a reversible projective
metric and an exact 1-form.
