What is \(\phi^*(f)\)?
First, we will think about \(-\phi^*(f)\) instead, which is more straightforward geometrically. Just ignore the minus sign for now and think of "\(-\phi^*(f)\)" as an atomic lexical unit.
We write \[-\phi^*(f) = \inf_{x\in E}\{\phi(x) - \langle f, x\rangle\},\] i.e. the quantity \(-\phi^*(f)\) is the lowest (i.e. "most negative") value of the signed vertical distance between the graph of \({\color{blue}{\phi}}\) and the graph of \({\color{red}{x\mapsto \langle f, x\rangle}}\).
Change the value of \(f\) with this slider to see how \(-\phi^*(f)\) is built geometrically.
\(f = ~\)
Note how for \(f > 0.7\), the direction of the arrow reverses and \(-\phi^*(f)\) becomes negative.
Later, when we talk about the Fenchel-Rockafellar Theorem, we will see a term of the form \(-\phi^*(-f)-\psi^*(f)\). In order to understand this quantity we will need a visualization of \(-\phi^*(-f)\) in a plot of \(-\phi\) (this will make more sense later when we talk about the Fenchel-Rockafellar theorem).
We see that \[-\phi^*(-f) = \inf_{x\in E} \{\langle f, x\rangle - (-\phi(x))\},\] which is the lowest value of the signed vertical distance between the graph of \({\color{red}{x\mapsto \langle f, x\rangle}}\) and the graph of \({\color{blue}{\phi}}\).
\(f = ~\)
We see that \(-\phi^*(-f)\) is (naturally) just mirrored at the ordinate.