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?



Derivability properties of the distance function in a Finsler Manifold
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 $d(0,x) = f(x)$.


