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?