# When is a function a convex conjugate?

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.

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 $$f(x) = \sup_{x^*\in X^*} \langle x^*,x\rangle - g(x^*).$$

• 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.
– gerw
Apr 13, 2021 at 17:37
• True - and luckily, that's exactly what was asked for… Thanks for the catch!
– Dirk
Apr 14, 2021 at 8:35