Buckley, S.M. and Falk, K. and Wraith, D.J.
Ptolemaic spaces and CAT (0).
Glasgow Mathematical Journal, 51.
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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