I was reading Villani's book: Optimal Transportation, Old and New.
From page 80-83, he introduced some results about dual formulation of transport inequality.
Assume $C(\mu,\nu)$ is the optimal transport distance from probability measure $\mu$ (defined on $\mathcal{X}$) to $\nu$ (defined on $\mathcal{Y}$), with cost function $c(\cdot,\cdot)$.
Given a convex functional $F(\cdot)$ defined on $P(\mathcal{X})$, we can define its Legendre Transformation : $L(F)=\Lambda$, so $\Lambda$ is a convex functional on $C_b(\mathcal{X})$.
The main result (Theorem 5.26) is:
\begin{equation*}
\forall \mu \in P(\mathcal{X}), C(\mu,\nu)\leq F(\mu)
\end{equation*}
and
\begin{equation*}
\forall \phi \in C_b(\mathcal{Y}), \Lambda(\int_\mathcal{Y}\phi d\nu-\phi^c)\leq 0\quad \phi^c:=\sup_{y\in\mathcal{Y}}(\phi(y)-c(x,y))
\end{equation*}
Are equivalent.
Well this result is used to analyze transport inequalities. In Example 5.29, the author gives an important application of this result:
Assume $\mathcal{Y}=\mathcal{X}$ and consider:
\begin{equation*}
F(\mu):=KL(\mu||\nu)=\int_{\mathcal{X}}\ln(\frac{d\mu}{d\nu})d\mu
\end{equation*}
which is the Kullback Liebller Divergence between measures (also known as relative entropy);
We could compute the Legendre Transformation of $F$, which has the following form:
\begin{equation*}
\Lambda(\phi):=\ln(\int_{\mathcal{X}}e^{\phi}d\nu)
\end{equation*}
Thus by the previous result, the transportation-entropy inequality
\begin{equation*}
C(\mu,\nu)\leq KL(\mu||\nu) \quad \forall \mu\in P(\mathcal{X}) \quad (1)
\end{equation*}
is equivalent to :
\begin{equation}
\ln(\int_{\mathcal{X}}e^{\int \phi d\nu-\phi^c}d\nu)\leq 0
\Leftrightarrow e^{\int_{\mathcal{X}}\phi d\nu}\leq (\int_{\mathcal{X}}e^{-\phi^c}d\nu)^{-1} \quad (2)
\end{equation}
But in the book, the author directly arrives at:
\begin{equation}
e^{\int_{\mathcal{X}}\phi d\nu}\leq \int_{\mathcal{X}}e^{\phi^c}d\nu \quad (3)
\end{equation}
If we apply Cauchy inequality, we will deduce from (1) to (3):
\begin{equation*}
e^{\int_{\mathcal{X}}\phi d\nu}\leq (\int_{\mathcal{X}}e^{-\phi^c}d\nu)^{-1}\leq \int_{\mathcal{X}}e^{\phi^c}d\nu
\end{equation*}
But how could we deduce from (3) back to (1)? I am quite confused about it. I even think that $(3)$ and $(1)$ are equivalent is not a trivial corollary from our previous Theorem 5.26.
Can any expert in this area help me with this problem? So many thanks!
2026-02-24 00:32:11.1771893131