The Statement of Kantorovitch Optimal Transport Problem vs Monge. Hi I'm looking for some intuition into Kantorovitch's statement of the Optimal Transport Problem. Specifically some help with interpretation of the diagrams https://speakerdeck.com/gpeyre/numerical-optimal-transport-1?slide=26. 
Monge's problem asks for the optimal (w.r.t some cost function) mapping $T$ which takes one measure $\alpha$ on $\mathcal{X}$ to another $\beta$ on $\mathcal{Y}$ via the push-forward $T_{\#}\alpha=\beta$. For me the map $T:\mathcal{X}\to \mathcal{Y}$ has a physical interpretation which tells us exactly how to transport mass, that is any mass at $x$ is transported to $T(x)$. 
A problem with Monge's formulation is that it does not allow for the splitting of mass. A relaxation of the problem is Kantorovitchs formulation which instead asks for a optimal measure $\pi$ on $\mathcal{X}\times \mathcal{Y}$ with marginals $\alpha$ and $\beta$. 
$Question : $
How should I physically interpret the mesaure/plan $\pi$? Specifically can someone describe the continuous diagram in the link provided? 
 A: You can view the coupling $\pi$ as a non-deterministic version of $T$, because functions cannot map the same value to different points.
One advantage of using an optimal coupling $\pi$ as transport plan is that it exists way more often than a map $T$, and if you have a $\pi$ which is an optimal transport from one measure $\mu$ to another measure $\nu$, then you always get an optimal transport from $\nu$ to $\mu$ as well - in the finite case when $\pi$ is a matrix that would be the transpose $\pi^T$.
The line you see in that image tells you how much mass gets transported from the density $\alpha$ to points of mass in the density $\beta$. I always imagine they are piles of sand, so you can push all the "grains of sand" from $\alpha$ horizontally until they hit the curve, and then push them up vertically to become $\beta$.
It might be useful to make a discrete example with distributions which look similar to $\alpha$ and $\beta$ and see how the matrix $\pi$ changes as you add more and more bins.
