Some inequalities in Ptolemaic spaces (Q656158)

A theorem of Ptolemy states that if \(A\), \(B\), \(C\), \(D\) are any points in the Euclidean plane, then \[ \| A-B\|~\| C-D\|+ \| A-D\|~ \| C-B\| \geq \| A-C\|~\| D-B\|, \] where \(\|\;\|\) is the Euclidean norm. The same inequality holds in any real inner product space, as proved by \textit{I. J. Schoenberg} in [Ann. Math., Princeton, (2) 41, 715--726 (1940; JFM 66.0898.02)]. Following \textit{M. S. Klamkin} and \textit{A. Meir} in [Pac. J. Math. 101, 389--392 (1982; Zbl 0499.46008)], the authors of the paper under review call a real normed space with norm \(\|\;\|\) \textit{Ptolemaic} if the inequality above holds for all points \(A\), \(B\), \(C\), \(D\). Then they establish several inequalities that must hold in such spaces. It would probably be interesting to characterize those Minkowski planes which are Ptolemaic in terms of the shapes of their unit balls.