The Geodesic Problem in Nearmetric 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 nearmetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzelà theorem) still hold in nearmetric spaces. Moreover, we explore conditions under which a nearmetric will induce an intrinsic metric. As an example, we introduce a family of nearmetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these nearmetrics coincide with the dα metric studied early in [6]. Moreover, optimal transport paths between atomic probability measures turn out to be geodesics in these intrinsic metric spaces.