Variational problems concerning length distances in metric spaces

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