Exercise 2.14 in Villani's "Topics in Optimal Transportation" asks us to show the following:
Let $Q=[0,1]^{n-1}$, and $\mu$ is uniform on $Q\times \{0\}$, $\nu$ uniform on $Q\times \{1\} \cup Q\times \{-1\}$, with the cost function $c(x,y)=|x-y|^2$. Then the only solution to the Kantorovich problem is $\pi=(\delta_{x_1=1}+\delta_{x_1=-1})/2$. Additionally, the Monge problem has no solution.
To make it a bit simpler, I take $n=3$. My questions are the following:
I am having trouble even understanding the way that $\pi$ is written. Intuitively, one just needs to split the mass at each point of $Q\times \{0\}$ and transfer half of it to $Q\times \{1\}$ and the other half to $Q\times \{-1\}$, or, visually, stack three unit squares on top of each other with a unit distance between them, then $\mu$'s mass is concentrated on the middle one, and each point just splits half the mass perpendicularly to both the upper and the lower unit square ( the union of which is the support of $\nu$). But I do not see how $\pi=(\delta_{x_1=1}+\delta_{x_1=-1})/2$ is supposed to mean what I just described. Why $x_1$? I simply cannot understand what he means here.
I am assuming we are to use Knott-Smith to show that the this is the unique solution to the Kantorovich problem, but I have no clue where to start, though this could also be related to the fact that I cannot understand how $\pi$ is represented here.
Any ideas and hints would be great!
Thank you very much in advance!
Kind regards!