2
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

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

Improve this answer
$\endgroup$
3
  • $\begingroup$ Right! I now feel silly about this question. Thank you. $\endgroup$
    – e.lipnowski
    Apr 4, 2021 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, 2021 at 17:37
  • $\begingroup$ True - and luckily, that's exactly what was asked for… Thanks for the catch! $\endgroup$
    – Dirk
    Apr 14, 2021 at 8:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.