I know that generally the geodesic distance $d\_x$ from a point $x$ in a Finsler space is not smooth ($C^\infty$). According to Shen, the square of it is just $C^1$ at $x$. Now I am wondering if there exists any number $n$ that makes $(d\_x)^n$ smooth in $x$?



Smoothness of some power of the geodesic distance in a Finsler geometry
Without more restrictions on the Finsler structure, the answer is 'no, there need not be any finite $n$ such that the $n$-th power of the distance function is smooth'. 


For example, consider the Finsler structure on the $xy$-plane whose norm is
$$
F(x,y;\dot x, \dot y) = \sqrt{\dot x^2+\dot y^2}+\sqrt{\dot x^2 + 2\dot y^2}.
$$
The distance from $p=(0,0)$ to the point $(x,y)$ is then 
$$
d\_p(x,y) = \sqrt{ x^2+ y^2}+\sqrt{x^2 + 2y^2},
$$
and no positive integer power of this function is smooth on a neighborhood of $(0,0)$ in the $xy$-plane.


