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.



When is a function a convex conjugate?
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*
$$
f(x) = \sup\_{x^\*\in X^\*} \langle x^\*,x\rangle - g(x^\*).
$$


