Asymmetry in Hilbert's fourth problem

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.