Global Approximation of Convex Functions by Differentiable Convex Functions on Banach Spaces

D. Azagra ; C. Mudarra

Journal of convex analysis, Tome 22 (2015) no. 4, pp. 1197-1205

Résumé

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.

Détail

Classification : 46B20, 52A99, 26B25, 41A30
Mots-clés : Approximation, convex function, differentiable function, Banach space