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

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

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