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Let $X$ be a Banach space; $X^*$ be its dual; and $g:X^*\to\mathbb R\cup\{\infty\}$ be a proper, convex, weak${}^*$-lower semicontinuous function with weak${}^*$-compact effective domain.

Question: Is there a known characterization of when $g$ is/isn't the convex conjugate of some (proper, convex, lower semicontinuous) $f:X\to\mathbb R\cup\{\infty\}$?

Failing that, I would love any sufficient conditions for $g$ to admit such a representation.

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If I recall correctly, every proper, convex, weak*-lower semi-continuous function is a conjugate. In fact, it is the conjugate of its pre-conjugate $$ f(x) = \sup_{x^*\in X^*} \langle x^*,x\rangle - g(x^*). $$

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  • $\begingroup$ Right! I now feel silly about this question. Thank you. $\endgroup$ Apr 4 '21 at 18:50
  • $\begingroup$ I don't think that this is true (unless $X$ is reflexive). The conjugate of the preconjugate is always weak-* lower semicontinuous. Hence, you have to assume at least that $g$ is weak-* lower semicontinuous. $\endgroup$
    – gerw
    Apr 13 '21 at 17:37
  • $\begingroup$ True - and luckily, that's exactly what was asked for… Thanks for the catch! $\endgroup$
    – Dirk
    Apr 14 '21 at 8:35

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