Triangle inequalities in path metric spaces

Abstract

We study side-lengths of triangles in path metric spaces. We prove that unless such a space $X$ is bounded, or quasi-isometric to ${ℝ}_{+}$ or to $ℝ$, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in $X$. We construct an example of a complete path metric space quasi-isometric to ${ℝ}^2$ for which every degenerate triangle has one side which is shorter than a certain uniform constant.