In this article, we will introduce monotone transport maps and plans in 1D.

In this article we will introduce the definition of Reproducing Kernel Hilbert Space (RKHS).

In this article, we will show some sufficient conditions for optimality and stability.

In this article, we will prove the duality result for Kantorovich problem.

In this article, we will introduce some results for $c$-concavity and cyclical monotonicity, which will be used to prove the duality result in the next article.

In this article, we will consider $X=Y=\Omega \subset \mathbb{R}^d$ with $\Omega$ compact and $c(x,y)=h(x-y)$ for $h$ strictly convex.

In this article, we will talk about the dual problem Kantorovich.

In this article, we will talk about the problems of Monge and Kantorovich and the corresponding existence of solutions.

In this article, we will introduce some important properties of the relative entropy, including lower semi-continuity, convexity, compactness of sublevel sets based on the Donsker-Varadhan variational formula.

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