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.