More on Riemann metrics in \(\mathbb R^2\) for which the lines are the shortest paths (Q1129744)

The author studies the parametric version of Hilbert's fourth problem in the context of Riemann metrics: Describing all Riemann metrics on the Euclidean plane \(\mathbb R^2\) for which the usual lines are the geodesics (called Hilbert metrics). A criterion is derived giving necessary and sufficient conditions for the geodesics of a Riemann metric defined on \(\mathbb R^2\) to be the usual straight lines. As an application of this criterion the author finds out the unique Hilbert metric in a class of isotropic Riemann metrics with isotropic orientation function.