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- W2732635847 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 well-known results in metric spaces (e.g. Ascoli-Arzel`{a} 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." @default.
- W2732635847 created "2017-07-14" @default.
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- W2732635847 date "2008-07-21" @default.
- W2732635847 modified "2023-09-26" @default.
- W2732635847 title "The geodesic problem in quasimetric spaces" @default.
- W2732635847 hasPublicationYear "2008" @default.
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