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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.

MSC:

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