PTOLEMAIC SPACES AND CAT(0)

2009/04/27 by Stephen M. Buckley, Kurt Falk, David J. Wraith · 2 citations
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Mathematics and Applications #Ptolemy's table of chords #Converse #Mathematics #Generality #Pure mathematics #Manifold (fluid mechanics) #Space (punctuation) #Metric (unit) #Inequality #Metric space #Mathematical analysis #Geometry #Computer science

paper · pdf · doi:10.1017/s0017089509004984

openalex publication_date 2009/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Abstract We consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

Citations

• Metric Structures for Riemannian and Non-Riemannian Spaces
• An Introduction to Riemann-Finsler Geometry
• A Course in Metric Geometry
• Metric Spaces of Non-Positive Curvature
• A Panoramic View of Riemannian Geometry

Cited by

• Metric projections versus non-positive curvature
• Ptolemy's inequality [wikipedia]