Geodesics in asymmetric metric spaces

In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of "length spaces" (also known as "path-metric spaces" or "intrinsic spaces"). In this paper we continue the analysis of asymmetric metric spaces. We propose possible definitions of completeness and (local) compactness. We define the geodesics using as admissible paths the class of run-continuous paths. We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic. We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.