Variational problems concerning sub-Finsler metrics in Carnot groups

This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of $Γ$-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.

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