convex conjugate $f^*$ is proper if both $f$ and $f^{**}$ are

Question

If $f$ and $f^{**}$ on $\mathbb R^d$ are proper functions where $f^*$ stands for the convex conjugate of $f$ why does that follow that $f^*$ is proper, too?

Thanks a lot...

Answer 1

You could argue by contradiction. Assume that $f^*$ is not proper. Then it is either (1) equal to the constant $+\infty$ function or (2) it attains the value $-\infty$ somewhere (say, at $y_0$).

Now look at the definition of $f^{**}$:

$$ f^{**}(x) = \sup_{y \in \mathbb R^d} [(x,y) - f^*(y)]$$

Can you see what consequences (1) and (2) have for properness of $f^{**}$, respectively?