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 as f ( x ) = max x ∗ ∈ 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 M f of X ∗ for which the above duality holds.