Triangle inequalities in path metric spaces

2006/11/06 by Michael Kapovich
Computer Science · Mathematics · #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #Injective metric space #Metric (unit) #Metric space #Optimization and Variational Analysis #Path (computing) #Space (punctuation) #Triangle inequality #math.DG #math.GR #math.MG #msc:51K05

paper · pdf · doi:10.2140/gt.2007.11.1653

published as Geom. Topol. 11 (2007) 1653-1680 · 21 pages, 6 figures

arxiv created 2006/11/06 · openalex publication_date 2007/08/02 · arxiv updated 2014/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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 C 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. 51K05 1 Introduction Given a metric space X define K 3 .X / WD f.a; b; c/ 2 3 C W there exist points x; y; z with d.x; y/ D a; d.y; z/ D b; d.z; x/ D cg:

Citations

• A Course in Metric Geometry
• On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds