The Geodesic Problem in Quasimetric Spaces

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 dα 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.