
Journal of Convex Analysis 22 (2015), No. 4, 11971205 Copyright Heldermann Verlag 2015 Global Approximation of Convex Functions by Differentiable Convex Functions on Banach Spaces Daniel Azagra ICMAT, Dep. de Análisis Matemático, Facultad Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain azagra@mat.ucm.es Carlos Mudarra ICMAT, Calle Nicolás Cabrera 1315, Campus de Cantoblanco, 28049 Madrid, Spain carlos.mudarra@icmat.es [Abstractpdf] We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every real number $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:U \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of {\em continuous} (not necessarily bounded on bounded sets) convex functions by $C^k$ smooth convex functions can be reduced to the problem of global approximation of {\em Lipschitz} convex functions by $C^k$ smooth convex functions. Keywords: Approximation, convex function, differentiable function, Banach space. MSC: 46B20, 52A99, 26B25, 41A30 [ Fulltextpdf (117 KB)] for subscribers only. 