# Optimal transport: how $\varphi^c$ can be written as $\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$?

Let $$X,Y$$ be Polish spaces and $$c:X \times Y \to [0, \infty]$$ lower semi-continuous. There is a sequence $$(c_\ell)_{\ell \in \mathbb N}$$ with $$c_\ell:X \times Y \to [0, \infty)$$ of bounded Lipschitz continuous functions such that $$c_\ell \uparrow c$$ pointwise. Fix $$\varphi:X \to \mathbb R$$. The following is taken from page 33 of Villani's Topics in Optimal Transportation.

Remark 1.12 (c-concave functions). It follows from the proof that, when $$c$$ is bounded, one can restrict the supremum in the right-hand side of (1.4) to those pairs $$\left(\varphi^{c c}, \varphi^c\right)$$, where $$\varphi$$ is bounded and $$(1.18) \quad \varphi^c(y)=\inf _{x \in X}[c(x, y)-\varphi(x)], \quad \varphi^{c c}(x)=\inf _{y \in Y}\left[c(x, y)-\varphi^c(y)\right]$$ An easy argument shows that $$\left(\varphi^{c c}\right)^c=\varphi^c$$ (see Exercise 2.35). The pair $$\left(\varphi^{c c}, \varphi^c\right)$$ is called a pair of conjugate $$c$$-concave functions. Note that $$\varphi^c$$ is measurable, since it can be written (exercise) as $$\lim _{\ell \rightarrow \infty} \psi_{\ell}$$, where $$\psi_{\ell}(y)=\inf _{x \in X}\left[c_{\ell}(x, y)-\varphi(x)\right],$$ and $$c_{\ell}$$ is an increasing family of bounded uniformly continuous functions converging pointwise to $$c$$. Indeed, each $$\psi_{\ell}$$ is uniformly continuous, and therefore $$\varphi^c$$ is measurable. Similarly, $$\varphi^{c c}$$ is measurable.

I proved that $$\psi_\ell$$ is bounded Lipschitz continuous for each $$\ell \in \mathbb N$$.

Could you explain how $$\varphi^c$$ can be written as $$\varphi^c = \lim _{\ell \rightarrow \infty} \psi_{\ell}$$?

I posted this question on MSE, but have not received any answer so far. So I post it here.

Are you sure there aren’t additional conditions on $$\varphi$$? Because otherwise taking $$X = \mathbb R$$ and $$Y$$ to be a one point space, the following gives a counterexample:

$$c(x, y) = 0$$ if $$x = 0$$; $$c(x, y) = 1$$ otherwise, and

$$\varphi(x) = 0$$ if $$x = 0$$, $$\varphi(x)= 2$$ otherwise.

Indeed, $$\varphi^c = -1$$, while $$\psi_\ell \leq -2$$ for all $$\ell$$.

My guess is that $$\varphi$$ should be restricted to be continuous.

• You are right! At the beginning of page 20, he said that "...Furthermore, it does not change the value of the supremum in the right-hand side of (1.4) if one restricts the definition of $\Phi_c$ to those functions $(\varphi, \psi)$ which are bounded and continuous". Nov 20, 2022 at 12:53
• I have a closely related question here. If you don't mind, please have a look at it. Nov 20, 2022 at 12:56
• Sure! I’ll have a look tomorrow morning, gotta go to bed soon now… Nov 20, 2022 at 12:57
• Thank you so much for your help. Good night! Nov 20, 2022 at 12:57