On the moduli of convexity
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- by A. J. Guirao and P. Hajek PDF
- Proc. Amer. Math. Soc. 135 (2007), 3233-3240 Request permission
Abstract:
It is known that, given a Banach space $(X,\|\cdot \|)$, 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.References
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Additional Information
- A. J. Guirao
- Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain
- Email: ajguirao@um.es
- P. Hajek
- Affiliation: Mathematical Institute, AV ČR, Žitná 25, 115 67 Praha 1, Czech Republic
- Email: hajek@math.cas.cz
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 3233-3240
- MSC (2000): Primary 46B03
- DOI: https://doi.org/10.1090/S0002-9939-07-09030-2
- MathSciNet review: 2322754