Global approximation of convex functions by differentiable convex functions on Banach spaces

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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 X, for every ε >0, and for every continuous and convex function f:U → R (not necessarily bounded on bounded sets) there exists a convex function g:X → R of class C¹(U) such that f-ε≤ g≤ f on U. We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by Cᵏ smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by Cᵏ smooth convex functions.