It is well-known that a convex function can be represented as the pointwise supremum of some affine functions. Hence let $f:\mathbb{R}\to \mathbb{R}$ be a convex function, we know there exists a set $\mathcal{A}$ such that $$ f(x)=\sup_{\alpha\in \mathcal{A}}a(\alpha)x+b(\alpha). $$
My question is, with suitable assumptions on $f$, is it possible to deduce certain properties of this representation? For example, $\mathcal{A}$ is compact, or $a$ and $b$ are continuous in $\alpha$?
you can use the fact that the double conjugate is the function itself (since your $f$ is convex and closed), so you can use $\mathcal{A} = \mathbb{R}$, $a(\alpha)=\alpha$ and $b(\alpha) = -f^*(\alpha)$. If your function is polyhedral, then so is $b$.