#### Volume 11, issue 3 (2007)

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Triangle inequalities in path metric spaces

### Michael Kapovich

Geometry & Topology 11 (2007) 1653–1680
 arXiv: math.MG/0611118
##### 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.

##### Keywords
path metric spaces, triangles
Primary: 51K05