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### Optimal transport for applied mathematicians: how does $\varphi (x) = \inf_{y \in Y} [c(x, y) - \psi (y)] \neq -\infty$ follow in Theorem 1.37?

I hope I did not misunderstand the question, but it seems $\varphi(x) > - \infty$ holds as follows if $(x, y) \in \Gamma$: For any $(x_i, y_i) \in \Gamma$, $i=1, \dots, n$, we see that \begin{...
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### For which value of $C(f)$ would the following inequality hold?

The question is not well posed, as it is not quite clear in what terms you want $C(f)$ to be expressed. If one only uses the terms you did mention -- the Lipschitz constant and the size of the domain, ...
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