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On the moduli of convexity

Authors: A. J. Guirao and P. Hajek
Journal: Proc. Amer. Math. Soc. 135 (2007), 3233-3240
MSC (2000): Primary 46B03
Published electronically: June 22, 2007
MathSciNet review: 2322754
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Abstract: It is known that, given a Banach space $ (X,\Vert\cdot\Vert)$, the modulus of convexity associated to this space $ \delta_X$ is a non-negative function, non-decreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation $ \frac{\delta_X(\varepsilon)}{\varepsilon^2}\leq 4L\frac{\delta_X(\mu)}{\mu^2}$ for every $ 0<\varepsilon\leq\mu\leq 2$, where $ L>0$ is a constant. We show that, given a function $ f$ satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to $ f$, in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.

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Additional Information

A. J. Guirao
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain

P. Hajek
Affiliation: Mathematical Institute, AV ČR, Žitná 25, 115 67 Praha 1, Czech Republic

Keywords: Banach spaces, modulus of convexity, uniformly rotund norms
Received by editor(s): June 26, 2006
Published electronically: June 22, 2007
Additional Notes: The first author was supported by grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain), and AP2003-4453 of MECD (Spain)
The second author was supported by the Institutional Research Plan, AV0Z10190503 and A100190502.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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