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

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.

Throughout this article $(X,d)$ will be a given complete metric space.

The Wasserstein distance has wide applications in problems such as optimal transport. It characterizes the distance between two probability measures. In German, Wasser means water, and Stein means stone (although I have not carefully verified whether Wasserstein is a person’s name; after all, many German names are of this kind, for example: Einstein = one stone). This article will mainly answer the following two questions:

• On what set is the Wasserstein distance defined?
• Why is the Wasserstein distance a distance? That is, why does it satisfy the three axioms of a metric?

In this article, we only consider the Wasserstein distance on the $d$-dimensional Euclidean space $\mathbb{R}^d$. Since $\mathbb{R}^d$ is a special Polish space, the conclusions of this article can also be extended to general Polish spaces.