How unique minima are guaranteed to exist in optimal transport. Why such a thing is not possible in general linear programming problems. What theorem guarantees the existence of unique minima to exist in optimal transport?
Edit: My question is regarding Kantorowich optimal transport
For a cost function $C(x,y)$ $$W[\pi]=\bigg [\sum C(x,y)\pi(x,y)\bigg]_{\text{inf}}$$ $$\sum_{y}\pi(x,y)=\rho(x)$$ $$\sum_{x}\pi(x,y)=\tilde{\rho}(y)$$
We seek for $\pi$ which minimizes the $W$