Generalized centers and characterizations of inner product spaces (Q1685890)

The aim of this article is to generalize results of \textit{J. Mendoza} and \textit{T. Pakhrou} [Math. Scand. 97, No. 1, 104--114 (2005; Zbl 1089.46017)] using absolute normalized norms on \(\mathbb{R}^{3}\). There is a natural generalization of the notion of \(p\)-centres of three point sets using absolute norms. In terms of these generalised centres, the authors show the same characterization of inner product spaces as Mendoza and Pakhrou [loc.\,cit.], for a certain class of absolute normalized norms on \(\mathbb{R}^{3}\) containing symmetric, strictly convex and smooth ones as well as the \(l_{p}\) norms for \(1 < p < \infty \). There are also new Garkavi-Klee type characterizations of inner product spaces using the notion of generalized centres of three point sets introduced by using absolute normalized norms.