What is an illustrative example of a Finsler manifold? I've attempted to get a bead on Finsler manifolds before attending an upcoming seminar that involves them. I've done some reading in the literature and think I understand that they are similar to Riemannian manifolds, but that the metric tensor need not be a quadratic differential and need not be isotropic. Also, the "sphere" of unit tangent vectors at a point in a Finsler manifold need not be symmetric although it does have to be convex.
I'd love to hear a nice, illustrative, concrete example of a Finsler manifold, especially if it could involve a picture and an explanation of why it's interesting to you.
Thanks!
 A: This is more about Finsler norms/metrics but hopefully it is still interesting.
An interesting one appears in the medical imaging literature, specifically for streamline tracking in the brain using diffusion imaging.
Essentially, the goal is to discern the paths of the neural tracts in the brain. One is given a (noisy) tensor field describing the local diffusion properties at each point in the tissue. 
A Finsler norm is defined based on a high order diffusion tensor $D$:
$$
F(x,y) = \sqrt[n]{D_{i_1,\ldots,i_n}(x) y^{i_1} \ldots y^{i_n} }
$$
Then the Finsler metric tensor is related to its Hessian:
$$
g_{ij}(x,y) = \frac{1}{2}\frac{\partial F^2(x,y)}{\partial y^i \partial y^j}
$$
The fact that the norm can depend on both orientation and spatial position is important here because streamlines can cross in the brain, making the use of a Riemannian metric alone unable to capture all the relevant information.
See Finsler Streamline Tracking with Single Tensor Orientation Distribution Function for High Angular Resolution Diffusion Imaging and 
Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion Imaging by Astola et al.
