Exponential Map for non-smooth Finsler manifolds

Context

If I'm interested in studying reversible Finsler manifolds which do not have the strong convexity of the Hessian property (that is the Finsler function is a regular norm on every tangent space). However they should have the property that there exists a unique minimizing geodesic between every two curves (unlike the $\ell^1$-metric).

Question:

Is the exponential map well-defined? If so will it be a homeomorphism onto its image?

• Between every two points you mean (?) The problem with defining the exponential, at least naively, is that your unique geodesic may not be differentiable. I've never thought about the necessary and sufficient conditions for a Finsler metric to have locally unique differentiable geodesics, but for locally unique geodesics Pogorelov showed that C1,1 and strictly convex are sufficient: see here: ams.org/mathscinet/search/… – alvarezpaiva Apr 25 '17 at 14:17
• Yes I am familiar with this result. However, the geodesics in $\ell^1$ need not have derivatives everywhere, so unfortunately it doesn't help me :/ – AIM_BLB Apr 25 '17 at 16:38
• Didn't you explicitly rule out $\ell_1$ in your question? The wording seems to indicate you are looking for the conditions for a Finsler metric to be a G-space. Is that a correct interpretation? – alvarezpaiva Apr 27 '17 at 16:57