Topological equivalence of some variational problem associated to distances

To every distance d on a given open set \Omega\subseteq\mathbb R^n, we may associate several kinds of variational problems. We show that, on the class of all geodesic distances d on \Omega which are bounded from above and from below by fixed multiples of the Euclidean one, the uniform convergence on compact sets turns out to be equivalent to the \Gamma-convergence of each of the corresponding variational problems under consideration