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## Triangle inequalities in path metric spaces

## Published Web Location

https://arxiv.org/pdf/math/0611118.pdfNo data is associated with this publication.

## 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 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.