# Characterization of convex dual-like functions

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 $$f(x)=\sup \left \{ \left. \left\langle x^{\star} , x \right\rangle - g\left( x^{\star} \right) \right| x^{\star} \in X^{\star} \right\} ?$$

• Any reason why you don't expect $g=f^*$? The double conjugate of a closed convex function is the function itself, after all. – LinAlg Oct 30 '18 at 0:02
• In the reflexive case it's obvious that's $g=f^{\star}$, but I saw (potentially eronious) example recently in the non reflexive case.. – AIM_BLB Oct 30 '18 at 1:29