Inserted: 28 oct 2013
Last Updated: 7 jan 2015
Journal: Analysis and Geometry in Metric Spaces
"Analysis and Geometry in Metric Spaces" is an open access paper: if interested, please download the paper from that source. The DOI link will bring you to the correct web page.
In a recent paper we studied "asymmetric metric spaces"; in this context we studied the length of paths, introduced the class of run-continuous paths; we noted that there are different definitions of ``length space'' (also known as ``path-metric space'' or ``intrinsic space'').
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 that 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.