Volume 11, issue 3 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 + or to , 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 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
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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/