On the moduli of convexity

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2007-10, Vol.135 (10), p.3233-3240
Main Authors: Guirao, A. J., Hájek, P.
Format: Article
Language:English
Subjects:
Applied mathematics
Banach space
Convexity
Coordinate systems
Equivalence relation
Exact sciences and technology
Functional analysis
General mathematics
General, history and biography
Increasing functions
Mathematical analysis
Mathematical constants
Mathematical functions
Mathematics
Sciences and techniques of general use
Triangles
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Summary: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 00 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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-07-09030-2