Variational problems concerning sub-Finsler metrics in Carnot groups

Abstract

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.