2009/02/07 by Qinglan Xia · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.1007/s12220-008-9065-4
openalex publication_date 2009/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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.