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Convex-transitive characterizations of Hilbert spaces

Published online by Cambridge University Press:  26 May 2009

Jarno Talponen
Jarno Talponen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Box 68, (Gustaf Hällströmin katu 2B) 00014University of Helsinki, Finland ([email protected])
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Abstract

We investigate real convex-transitive Banach spaces X, which admit a one-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. For example, if uSX is a big point such that there is a bicontractive linear projection P : X → [u] and X* is weak*-locally uniformly rotund, then X is a Hilbert space. The results obtained here are motivated by the well-known Banach—Mazur rotation problem, as well as a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 3 , June 2009 , pp. 633 - 659
Copyright
Copyright © Royal Society of Edinburgh 2009

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