Generalized centers and characterizations of inner product spaces. (English) Zbl 1386.46026

The aim of this article is to generalize results of J. Mendoza and 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.


46C15 Characterizations of Hilbert spaces
46B20 Geometry and structure of normed linear spaces
Full Text: Euclid