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Uniformly convex functions on Banach spaces. (English) Zbl 1184.52009
Authors’ summary: 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.

##### MSC:
 52A41 Convex functions and convex programs in convex geometry 46G05 Derivatives of functions in infinite-dimensional spaces 46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics 49J50 Fréchet and Gateaux differentiability in optimization 90C25 Convex programming
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