Geodesic
PTOLEMAIC SPACES AND CAT(0)
Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 301-314

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DOI

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.
DOI : 10.1017/S0017089509004984
Mots-clés : 53C20, 53C60, 51F99 
BUCKLEY, S. M.; FALK, K.; WRAITH, D. J. PTOLEMAIC SPACES AND CAT(0). Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 301-314. doi: 10.1017/S0017089509004984
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     author = {BUCKLEY, S. M. and FALK, K. and WRAITH, D. J.},
     title = {PTOLEMAIC {SPACES} {AND} {CAT(0)}},
     journal = {Glasgow mathematical journal},
     pages = {301--314},
     year = {2009},
     volume = {51},
     number = {2},
     doi = {10.1017/S0017089509004984},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089509004984/}
}
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