Q:

Derivability properties of the distance function in a Finsler Manifold

We know that, in a Riemannian manifold, the geodesic distance between a point O and a point P, when we fix O, is a function of P that is $C^\infty$ everywhere on a local neighborhood, except in P=O. If we consider the squarred function, we have $C^\infty$ everywhere. 
(There is a topic on that subject)
Now, in Finsler's work in 1918, and in Riemann's habilitation thesis in 1854, the authors consider a Finsler's manifold of a particular kind.  They consider that the metric is given by the $(2n)^{th}$ root of an homogeneous form of degree $2n$:
$ds (X, dx)=^{2n}\sqrt{g_{\mu_1, \cdots, \mu_{2n}}(X).dx^{\mu_1}\cdots dx^{\mu_{2n}}}$.
My question is:
What can we say, in this framework, about the differentiability properties of the geodesic distance, and of the $2n^{th}$ power of the geodesic distance? 

A:

I believe that this is discussed in Zhongmin Shen's book "Lectures on Finsler Geometry".   The result, if I remember correctly, is that the square of the geodesic distance function is generally not smooth (i.e., infinitely differentiable) at the origin in the Finsler case (as opposed to the Riemannian case), though it is smooth in a punctured neighborhood of the origin. 
You can see this easily in the case of a Finsler metric on $\mathbb{R}^n$ that is translation invariant, i.e., the Finsler function is $F(x,\dot x) = f(\dot x)$ where $f:\mathbb{R}^n\to\mathbb{R}$ has the properties that $f$ is homogeneous of degree $1$ and smooth away from $0\in\mathbb{R}^n$ and that $f^2$ is strictly convex away from the origin but not smooth there.   Then the geodesic distance function from the origin in standard coordinates is just $d(0,x) = f(x)$.

