This is my first time posting a thread. I apologize if I somehow do not comply with the rules (please remind me if it happens, so next time I can do it correctly:)
Today I was having an optimization class, and had a question which my professor left unexplained.
Suppose f is a convex function from R^n to R (including infinity), and its conjugate is lower unbounded, then f is always infinite. I cannot see why, and I search online and textbook, but to little avail.
I would really appreciate it if someone could explain that a bit. Thank you!
This statement is false. Let $ b\in \mathbb R^n $ be nonzero and let $f $ be the indicator function of the set $ S=\{b\}$. Then $ f^*(z) = \langle z, b\rangle$. So $ f^*$ is unbounded below, but $ f(b)$ is finite.
(I'm using the term "indicator function" in the convex analysis sense.)