Variational problems concerning length distances in metric spaces

Given a locally compact, complete metric space $({\rm X},{\sf D})$ and an open set $\Omega\subseteq{\rm X}$, we study the class of length distances $\sf d$ on $\Omega$ that are bounded from above and below by fixed multiples of the ambient distance $\sf D$. More precisely, we prove that the uniform convergence on compact sets of distances in this class is equivalent to the $\Gamma$-convergence of several associated variational problems. Along the way, we fix some oversights appearing in the previous literature.