Villiani writes (some notation changed) in Topics in Optimal Mass Transportation:



> 
> Theorem 1.9. Let $E$ be a normed VS, $E^\*$ it topological dual. $\Theta$ and $\Psi$ are two convex functions on $E$ with values in $\mathbb{R}\cup{+\infty}$. Let $\Theta^\*$ and $\Psi^\*$ be Legendre-Fenchel transforms of $\Theta$ and $\Psi$ (Earlier definited as $R^\*(z^\*) = \sup\_{z\in E}[<z^\*,z>-R(z)]$). Assume that there exists $z\_0\in E$ such that $\Theta(z\_0)$ and $\Psi(z\_0)$ are finite and $\Theta$ is continuous at $z\_0$. Then:
>  $$\inf\_E[\Theta + \Psi]=\max\_{z^\*\in E^\*}[-\Theta^\*(-z^\*)-\Psi^\*(z^\*)]$$
> 
> 
> 


He begins his proof by saying: We want to prove:
$$\sup\_{z^\*\in E^\*} \inf\_{x,y\in E} [\Theta(x) + \Psi(y) + <z^\*,x-y>]=\inf\_{x\in E}[\Theta + \Psi]$$
Then he says: The choice of $x=y$ shows that the LHS is not larger than the RHS.


1. I'm confused about where his "sup inf" expression is coming from. I mean I get there's a substitution but that's it. Does someone understand where this expression is coming from?
2. Doesn't a choice of $x=y$ establish equality?



Fenchel-Rockafellar Duality in Villani's Book
More details based on Steve's comment: We have
\begin{align}
 -\Theta^\*(-z^\*) &= - \sup\_{x \in E} \big[ \langle-z^\*,x \rangle - \Theta(x) \big] \\
 &=\inf\_{x \in E} \big[ \langle z^\*,x \rangle + \Theta(x) \big]
\end{align}
and
\begin{align}
 -\Psi^\*(z^\*) &= - \sup\_{y \in E} \big[ \langle z^\*,y \rangle - \Psi(y) \big] \\
 &=\inf\_{y \in E} \big[ \langle z^\*,- y \rangle + \Psi(y) \big]
\end{align}
Then,
\begin{align}
 -\Theta^\*(-z^\*) -\Psi^\*(z^\*) &= \inf\_{x,y \in E} \big[ \langle z^\*,x -y\rangle + \Theta(x) + \Psi(y)\big]
\end{align}


