Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Geodesics in asymmetric metric spaces

created by mennucci on 28 Oct 2013
modified on 07 Jan 2015


Published Paper

Inserted: 28 oct 2013
Last Updated: 7 jan 2015

Journal: Analysis and Geometry in Metric Spaces
Volume: 2
Pages: 115-153
Year: 2014
Doi: 10.2478/agms-2014-0004

"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.