It is well known that every point of a Riemannian manifold $(M,g)$ possesses a fundamental system $\{U_n\}_{n\in\mathbb N}$ of geodesically convex neighborhoods. This means that every pair of points in a neighborhood $U_n$ are joined by a (unique) geodesic that is entirely contained in $U_n$.
My question is whether or not the same statement holds in Finsler manifolds $(M,F)$. My notion of Finsler manifold is the general one coming from convex Hamiltonian dynamics: a Finsler metric $F:TM\to \mathbb [0,\infty)$ is a continuous function, smooth and positive outside the zero section, that is positively homogeneous of degree 1 (i.e. $F(\lambda v)=\lambda F(v)$ for all $\lambda>0$ and $v\in TM$) and fiberwise strongly convex (i.e. the restriction of $F$ to any fiber of $TM$ has positive definite Hessian at every point other than the origin).