PTOLEMAIC SPACES AND CAT(0)

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). Ptolem...

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Bibliographic Details
Published in:Glasgow mathematical journal 2009-05, Vol.51 (2), p.301-314
Main Authors: BUCKLEY, S. M., FALK, K., WRAITH, D. J.
Format: Article
Language:English
Subjects:
51F99
53C20
53C60
Mathematics
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Summary: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.
ISSN:0017-0895
1469-509X
DOI:10.1017/S0017089509004984