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PTOLEMAIC SPACES AND CAT(0)

Published online by Cambridge University Press:  01 May 2009

S. M. BUCKLEY ,
K. FALK  and
D. J. WRAITH
S. M. BUCKLEY
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Co. Kildare, Ireland e-mail: stephen.m.buckley@gmail.com; kurt.falk@nuim.ie; david.wraith@nuim.ie
K. FALK
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Co. Kildare, Ireland e-mail: stephen.m.buckley@gmail.com; kurt.falk@nuim.ie; david.wraith@nuim.ie
D. J. WRAITH
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Co. Kildare, Ireland e-mail: stephen.m.buckley@gmail.com; kurt.falk@nuim.ie; david.wraith@nuim.ie
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Abstract

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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.

Keywords

53C2053C6051F99
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 301 - 314
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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