Related to this post, but slightly altered.
Let $\mu$ uniform on $\{0\}\times [0,1]$, $\nu$ uniform on $\{-1,1\}\times [0,1]$. Consider the Kantorovich optimal transport problem of $\mu$ to $\nu$ with the squared distance cost function. Intuitively, the optimal transport should split the mass at each point of $\mu$ and transport it along a straight line, giving half the mass to the $-1$ part and the other half to the $1$ part of $\nu$ (see drawing).
According to the Knott-Smith Optimality Criterion, $\pi$ being optimal is equivalent to the support of $\pi$ being contained in the graph of the subdifferential of a convex lower semi-continuous function $\varphi$.
Now the support of $\pi$ is given by: $\{((0,a),(\pm 1, a))|a\in[0,1]\}$. So now I want to find a $\varphi$ convex l.s.c. such that this support is in its subdifferential. My lecture notes claim that $\varphi(x_1,x_2)=|x_1|$ should work, however I get that the graph of its subdifferential is: $\{((0,x_2),(a,0))|x_2\in\mathbb{R},a\in[-1,1]\}\cup\{((x_1,x_2),(\text{sgn}(x_1),0))|x_1 \in \mathbb{R}\setminus \{0\},x_2\in\mathbb{R} \}$.
So it seems like $\text{Supp}(\pi)\subseteq \text{Graph}(\partial \varphi)$ does not hold. This either means that my support is wrong, my subdifferential is wrong, or that my function $\varphi$ is wrong.
Any help and/or ideas would be greatly appreciated!
I think that your function $\varphi$ is wrong, $$ \varphi(x_1, x_2) = |x_1| + \frac12 x_2^2 $$ should work.