Triangle inequalities in path metric spaces

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

Mathematical Subject Classification 2000

Primary: 51K05

References

Forward citations

Publication

Received: 6 December 2006

Accepted: 30 July 2007

Published: 2 August 2007

Proposed: Walter Neumann

Seconded: Yasha Eliashberg, Martin Bridson

Authors

Michael Kapovich
Department of Mathematics University of California, Davis Davis CA 95616 USA
http://www.math.ucdavis.edu/~kapovich/

Volume 11, issue 3 (2007)

Michael Kapovich

Abstract