The geodesic problem in quasimetric spaces
Abstract
In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\leq \sigma (d(x,z)+d(z,y))$ for some constant $\sigma \geq 1$, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many wellknown results in metric spaces (e.g. AscoliArzelà theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the $d_{\alpha}$ metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a "tree shaped" branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces.
 Publication:

arXiv eprints
 Pub Date:
 July 2008
 arXiv:
 arXiv:0807.3377
 Bibcode:
 2008arXiv0807.3377X
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Differential Geometry;
 Mathematics  Functional Analysis;
 Mathematics  Optimization and Control;
 54E25;
 51F99;
 49Q20 (Primary);
 90B18 (Secondary)
 EPrint:
 21 pages, 5 figures, published version