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Topological equivalence of some variational problems involving distances

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  • 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.
    Mathematics Subject Classification: 49J45, 53C60.

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