Proposition [lcc-002H]

Let (X,A,L) be a minmax problem. Suppose A is a convex subspace of a vector space V, and L(x,-): A \to \mathbb {R} is affine for each x. Then there exists functions f: X \to \mathbb {R}, g: X \to V^*, so that L(x,a) = f(x)+\langle g(x),a \rangle .