Characterization of inner product spaces in V- and L-spaces
``V space'' is the space M satisfying the following two conditions (i) and (ii): (i) For each p,q,r,s\(\in M\) with betweennessese pqr and pqs, there exists a point \(T\in M\) such that betweennesses pqt, qrt and qst hold. (ii) For each p,q,r\(\in M\) there exist points \(m,s\in M\) satisfying the following three conditions: (a) m is a metric midpoint of q and r; (b) m is a metric midpoint of p and s; (c) \(pq=rs.\) M is an ``L space'' if it satisfies the following L property: If f is a foot of different two points p and q contained in a metric line L, then \(f\in L\) and it is also a foot of any point on any metric line L(p,q) containing p and q. The V spaces include Banach spaces, hyperbolic spaces and other metric spaces, and the L spaces include hyperbolic spaces and strictly convex Banach spaces. The authors also give the following property \(B^*\): Let p,q and r be arbitrary three points in M. If \(pq=pr\) holds, if p,q and r do not lie on a metric line, and if m is a metric midpoint of q and r, then there exists a point s on a metric ray from p to m satisfying the three equations (\(\alpha)\)-(\(\gamma)\): (\(\alpha)\) \(ps=pq=pr\); (\(\beta)\) \(ps=rs\); and (\(\gamma)\) \((qr)^ 2\cdot (pq)^ 2=4(pq)^ 2(qs)^ 2-(qs)^ 4\). They prove the main Theorem: An L space (or a V space) is a real inner product space if and only if it satisfies the property \(B^*.\) The essential part of their proof is to derive the congruence to \(E_ 2\).
scientific article; zbMATH DE number 4080148