Let $f$ be a proper, convex, lsc functional from, a locally-convex topological vector space, $X$ to $\mathbb{R}$. Then by Fenchel-Monreau Theorem, $$ f(x) =\sup \left \{ \left. \left\langle x^{\star} , x \right\rangle - f^{\star} \left( x^{\star} \right) \right| x^{\star} \in X^{\star} \right\} $$ where the convex-conjugate $f^{\star}$ is defined by $$ f^{\star}\left( x^{\star} \right) := \sup \left \{ \left. \left\langle x^{\star} , x \right\rangle - f \left( x \right) \right| x \in X \right\} . $$
Is there a characterization of the proper, lsc, convex functionals $g:X^{\star}\rightarrow \mathbb{R}$ such that $$ f(x)=\sup \left \{ \left. \left\langle x^{\star} , x \right\rangle - g\left( x^{\star} \right) \right| x^{\star} \in X^{\star} \right\} ? $$