Smoothness of some power of the geodesic distance in a Finsler geometry

Question

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 $x$?

Answer 1

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.