Journal of Convex Analysis 22 (2015), No. 4, 1197--1205
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 13-15, Campus de Cantoblanco, 28049 Madrid, Spain
carlos.mudarra@icmat.es
[Abstract-pdf]
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
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