As I mentioned in this question([Relations between distance function and gradient on a Riemannian manifold](https://math.stackexchange.com/questions/2739650/relations-between-distance-function-and-gradient-on-a-riemannian-manifold)), there's question says $d(x,y) = \sup \{f(x)-f(y):f \in C^{\infty}(M)$, with $\|\nabla f\| \leq 1\}$.

@Frieder Jäckel says he had a solution to the non-trivial part of the inequation, but too long to post it in a comment. So I am to ask it again it this question.