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On uniformly convex functions and uniformly smooth functions. (English) Zbl 0835.46067
Summary: We study uniformly convex functions and uniformly smooth functions in the framework of nonstandard analysis. We show that in a Banach space, a proper, lower semicontinuous and convex function $f$ is uniformly convex on the whole space if and only if its conjugate function is uniformly Fréchet differentiable on $R(\partial f)$. Let $\varphi: [0, \infty)\to (- \infty, \infty]$ be a function. We characterize the uniform convexity and the uniform smoothness of the function $x\mapsto \varphi(|x|)$ on bounded balls in a normed linear space. We also show sufficient conditions which ensure the uniform convexity and the uniform smoothness of the function $x\mapsto \varphi(|x|)$ on a whole normed linear space.
46S20Nonstandard functional analysis
46N10Applications of functional analysis in optimization and programming
49J50Fréchet and Gateaux differentiability