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)aff(C) is dense in X  , and (iii) f:C→Rf: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∈Cx∈C, where f∗f∗ denotes the Fenchel conjugate of f   and M=X∗M=X∗. We show that, under assumptions (i)–(iii), there is a unique minimal weak∗-closed subset MfMf of X∗X∗ for which the above duality holds.