Nonpositive Curvature and the Ptolemy Inequality

By T. Foertsch, A. Lytchak, V. Schroeder. 2007

71 citations recorded by the source corpus.

We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy.

Subfields: Geometry and Topology, Mathematical Physics, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/30d6339f2b444e6996377db9ca9f9a4844d5c037

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. I. Berg
2. I. Nikolaev
3. T. Foertsch
4. V. Schroeder
5. P. Auscher
6. Tuomas Hytönen

In Mathematical Physics

1. I. Berg
2. I. Nikolaev
3. T. Foertsch
4. V. Schroeder
5. B. A. Kakavandi
6. Christof Schötz

In Theoretical Computer Science

1. I. Berg
2. I. Nikolaev

Citations in the corpus

• Quasilinearization and curvature of Aleksandrov spaces (2008, methodology): “Remark 25 After submitting this paper the authors learned that Foertsch, Lytchak and Schroeder independently proved that each Aleksandrov � 0 domain is Ptolemaic [ 18 ].”
• Hyperbolicity, CAT(−1)-spaces and the Ptolemy inequality (2006, background): “The relation between nonpositive curvature and the Ptolemy inequality is studied in [ 10 ].”