Triangle inequalities in path metric spaces
Metric Geometry
2014-11-11 v1 Differential Geometry
Group Theory
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.
Cite
@article{arxiv.math/0611118,
title = {Triangle inequalities in path metric spaces},
author = {Michael Kapovich},
journal= {arXiv preprint arXiv:math/0611118},
year = {2014}
}
Comments
21 pages, 6 figures