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Here is an example which shows that also the continuity of $\varphi$ does not help. Let $X = \mathbb R$ and $Y = \{0\}$. In the sequel, we will just drop the $Y$-argument. Let $c \equiv 1$, $\varphi \ …

Set $G := \{(x,y) \in X \times Y : y = T(x)\}$. I think for $\gamma = \gamma_T$ we just have \begin{align*} \gamma(G) = \int_{X \times Y} \chi_G(x,y) \,\mathrm{d}\gamma(x,y) = \int_X \chi_G \mathbin\c …

The minimizers cannot lie on the boundary. In fact, denote by $E_i \subset E$ the set of all points which are transported to $x_i^*$. Then, $x_i^*$ has to be the center of mass of $E_i$ and, thus, can …

Here are some ideas, but I did not have the time to check every detail. Let $T \subset [0,1]$ be countable and dense. Using a diagonal sequence argument, one can find a subsequence such that $$ \mu_n^ …

I was also struggling with Exercise 2.36... I think that I am now able to solve it, although it seems that it is more difficult than it appears... The key seems to be the following theorem. Theorem: …