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Uniformly convex functions on Banach spaces

Authors: J. Borwein, A. J. Guirao, P. Hájek and J. Vanderwerff
Journal: Proc. Amer. Math. Soc. 137 (2009), 1081-1091
MSC (2000): Primary 52A41, 46G05, 46N10, 49J50, 90C25
Published electronically: October 3, 2008
MathSciNet review: 2457450
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Abstract: Given a Banach space ($ X$,$ \Vert\cdot\Vert$), we study the connection between uniformly convex functions $ f:X \to \mathbb{R}$ bounded above by $ \Vert\cdot\Vert^p$ and the existence of norms on $ X$ with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function $ f:X \to \mathbb{R}$ bounded above by $ \Vert\cdot\Vert^2$ if and only if $ X$ admits an equivalent norm with modulus of convexity of power type 2.

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Additional Information

J. Borwein
Affiliation: Faculty of Computer Science, Dalhousie University, Halifax, Nova Scotia B3H 1W5, Canada – and – School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, New South Wales 2308, Australia

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

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

J. Vanderwerff
Affiliation: Department of Mathematics, La Sierra University, Riverside, California 92515

Keywords: Convex function, uniformly smooth, uniformly convex, superreflexive.
Received by editor(s): March 16, 2007
Received by editor(s) in revised form: April 26, 2008
Published electronically: October 3, 2008
Additional Notes: The first author’s research was supported by NSERC and the Canada Research Chair Program.
The second author’s research was supported by the grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain) and AP2003-4453 of MECD (Spain).
The third author’s research was supported by the grants A100190502, IAA 100190801 and Inst. Research Plan AV0Z10190503.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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