Uniformly convex functions on Banach spaces

Borwein, J.; Guirao, A.; Hájek, P.; Vanderwerff, J.

doi:10.1090/S0002-9939-08-09630-5

Uniformly convex functions on Banach spaces
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by J. Borwein, A. J. Guirao, P. Hájek and J. Vanderwerff

Proc. Amer. Math. Soc. 137 (2009), 1081-1091

DOI: https://doi.org/10.1090/S0002-9939-08-09630-5

Published electronically: October 3, 2008

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Abstract:

Given a Banach space ($X$,$\|\cdot \|$), we study the connection between uniformly convex functions $f:X \to \mathbb {R}$ bounded above by $\|\cdot \|^p$ 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 $\|\cdot \|^2$ if and only if $X$ admits an equivalent norm with modulus of convexity of power type 2.

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