2009/06/10 by Stephen M. Buckley, S. M. Buckley, David J. Wraith +3 · 2 citations
Mathematics · #Abstract simplicial complex #Artificial intelligence #Class (philosophy) #Computer science #Convex metric space #Euclidean space #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Injective metric space #Intrinsic metric #Mathematics #Metric (unit) #Metric space #Ptolemy's table of chords #Pure mathematics #Simplicial complex #math.MG #msc:51F99
paper · pdf · doi:10.1017/s0305004110000125
published in Mathematical Proceedings of the Cambridge Philosophical Society 149(1), 93-104 (Cambridge University Press) · 13 pages
arxiv created 2009/06/10 · openalex publication_date 2010/05/10 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial complexes, we characterize those which are isometric to Euclidean space in terms of metric inversions.