The geodesic problem in quasimetric spaces

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dc.creatorXia, Qinglan
dc.date2008-07-22
dc.date2009-05-27
dc.date.accessioned2026-07-07T13:18:02Z
dc.date.available2026-07-07T13:18:02Z
dc.descriptionIn this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\leq σ(d(x,z)+d(z,y))$ for some constant $σ\geq 1$, 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.
dc.description21 pages, 5 figures, published version
dc.identifierhttps://arxiv.org/abs/0807.3377
dc.identifierhttp://arxiv.org/abs/0807.3377
dc.identifierJournal of Geometric Analysis: Volume 19, Issue2 (2009), Page 452-479
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/231303
dc.subjectMetric Geometry
dc.subjectDifferential Geometry
dc.subjectFunctional Analysis
dc.subjectOptimization and Control
dc.subject54E25, 51F99, 49Q20 (Primary), 90B18 (Secondary)
dc.titleThe geodesic problem in quasimetric spaces
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