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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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