# Geodesics of non-smooth Finsler structure, or non-smooth Lagrange problem

I need to find the geodesics of a certain Finsler structure on $\mathbb R^n$. The structure is determined by quite nice $\ell^1$-like norms on tangent spaces, so that it is reversible. However the problem is that the unit balls of these norms are polyhedral, and the usual uniform convexity condition is not satisfied. I am sure that this situation must have already been considered in the literature and am looking for appropriate references.