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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).

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    $\begingroup$ I think this is treated in Zhongmin Shen's book Lectures on Finsler Geometry, or maybe the Bao-Chern-Shen book Introduction to Riemann-Finsler Geometry. (I don't have these books handy, or I would check.) I believe that there is also a paper by J. H. C. Whitehead, Convex regions in the geometry of paths (Quarterly Jour. Math. Oxford, Series 3 (1932), 33-42) that might be useful, but it has been a long time since I looked at this, so I'm not sure. $\endgroup$ Oct 10, 2015 at 9:51
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    $\begingroup$ Yes, Whitehead is the first reference. This has nothing to do with Finsler geometry it's true for second-order differential equations and path geometries. $\endgroup$ Oct 20, 2015 at 8:09

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