The fenchel conjugate as described in detail here is
$$f^*(y)= \sup_{x \in \operatorname{dom} f } (y^Tx-f(x))$$
The above post linked some intuitions on what values the Fenchel conjugate obtains at individual inputs: it can interpreted as the smallest difference between the function and the function with a "slope" of $y$.
Now what about a geometric intuition on the graph of the function $f^{*}: \mathbb{R}^{n} \to \mathbb{R}$. What does it look like in general? For a closed convex function, I know $f^{* *} = f$, but besides pushing symbols, and "doing reverse gradient twice", is there a geometric intuition for why this would hold?
In other places, such as in Fenchel duality theorem, $f^{*}$ also appears, so I felt a geometric intuition on the graph might be interesting.
