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

We show that if

is a Banach space whose dual

X^{*} X ∗

has an equivalent locally uniformly rotund (LUR) norm, then for every open convex

U\subseteq X U ⊆ X

, for every real number

\varepsilon >0 ε > 0

, and for every continuous and convex function

f:U \rightarrow \mathbb{R} f : U → R

(not necessarily bounded on bounded sets) there exists a convex function

g:U \rightarrow \mathbb{R} g : U → R

of class

C^1(U) C 1 ( U )

such that

f-\varepsilon\leq g\leq f f − ε ≤ g ≤ f

on

U. U .

We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) convex functions by

C^k C k

smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by

C^k C k

smooth convex functions.