Geodesic
On Asymmetric Distances
Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 200-231.

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In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.
Mots-clés : Asymmetric metric, general metric, quasi metric, ostensible metric, Finsler metric, run–continuity, intrinsic metric, pathmetric, length structure 
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Mennucci, Andrea C.G. On Asymmetric Distances. Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 200-231. http://geodesic.mathdoc.fr/item/AGMS_2013_1_1_a9/