Ptolemaic spaces and CAT (0)
Buckley, S.M. and Falk, K. and Wraith, D.J. (2009) Ptolemaic spaces and CAT (0). Glasgow Mathematical Journal, 51 . pp. 301-314. ISSN 0017-0895
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.
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