The unique minimal dual representation of a convex function
Suppose (i) X is a separable Banach space, (ii) C is a convex subset of X that is a Baire space (when endowed with the relative topology) such that aff(C) is dense in X, and (iii) f:C→R is locally Lipschitz continuous and convex. The Fenchel-Moreau duality can be stated asf(x)=maxx*∈M[〈x,x*〉-f*(x*)], for all x∈C, where f* denotes the Fenchel conjugate of f and M=X*. We show that, under assumptions (i)-(iii), there is a unique minimal weak*-closed subset Mf of X* for which the above duality holds. © 2010 Elsevier Inc.
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