Journal of Geometric Analysis

, Volume 19, Issue 2, pp 452–479

The Geodesic Problem in Quasimetric Spaces

Authors

    • Department of MathematicsUniversity of California at Davis
Open AccessArticle

DOI: 10.1007/s12220-008-9065-4

Cite this article as:
Xia, Q. J Geom Anal (2009) 19: 452. doi:10.1007/s12220-008-9065-4

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)≤σ(d(x,z)+d(z,y)) for some constant σ≥1, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà 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 α 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.

Keywords

Optimal transport path Quasimetric Geodesic distance Branching structure

Mathematics Subject Classification (2000)

54E25 51F99 49Q20 90B18

Copyright information

© Mathematica Josephina, Inc. 2009