Triangle inequalities in path metric spaces

Details Triangle inequalities in path metric spaces Triangle inequalities in path metric spaces

2006-11-06

arxiv.org/abs/math/0611118v1

Mathematics

Metric Geometry

21 pages, 6 figures

Scientific paper

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, 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 the Euclidean plane, for which every degenerate triangle has one side which is shorter than a certain uniform constant.

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