Four-point characterizations of real inner product spaces (Q1396033)

A metric space \(M\) (in which we write \(xy\) for the distance from \(x\) to \(y\), and \(xyz\) for the metric betweenness relation (\(x\neq y\neq z\) and \(xy+yz=xz\))) is: (i) {convex}; (ii) {externally convex} if for all \(p\), \(q\) in \(M\) with \(p\neq q\), there exists \(x\in M\) different from \(p\) and \(q\), such that (i) \(px+qx=pq\); (ii) \(pq+qx=px\). It satisfies (P\(_1\)) (or (P\(_4\))) if \(pq=pr=rs\wedge qrs\Rightarrow \text{Emb}(pqrs)\) (or \(pq=pr=rs \wedge (rqs\vee rsq)\Rightarrow \text{Emb}(pqrs))\), where \(\text{Emb}(pqrs)\) stands for ``\(p, q, r, s\) are embeddable in the Euclidean plane''. The authors show that a complete, convex, externally convex metric space is an inner product space if it satisfies (P\(_1\)) or if it satisfies (P\(_4\)).