Upper Bound on the Wasserstein Distance I'm interested to know if it's possible to construct an upper bound on the Wasserstein distance in terms of the Kolgomorov distance. 
The Wasserstein distance can we written as
$$W_{1}\left(F, G\right)=\int_{-\infty}^{\infty}\left|F(x)-G(x)\right| d x$$
for distribution functions $F$ and $G$.
suppose that I have access to an upper bound on the Kolgomorov distance, ie
$$\sup_{x \in \mathbb{R}}|F(x) - G(x)| \le K$$
for some $K > 0$.
Is it then possible to find a found a bound on $W_{1}\left(F, G\right)$?
 A: There are no effective bounds except when $K=0.$  (When $K=0,$ $F=G$ everywhere and both distances are necessarily $0.$)
Denote the Kolmogorov metric by $|\quad|_\infty$ and the Wasserstein metric by $|\quad|_1.$  (This is standard notation in mathematics.)
Let $X$ be a random variable with a Bernoulli$(K)$ distribution and set $Y(\epsilon)=(1+\epsilon)X$ for any $\epsilon\ge 0.$   Let $F$ be the distribution function (CDF) of $X$ and $G(\epsilon)$ the distribution function of $Y(\epsilon).$ 
$F$ and $G(\epsilon)$ differ only on the interval $[1,1+\epsilon)$ where the former equals $1$ and the latter equals $1-K.$  Thus
$$|F-G(\epsilon)|_\infty = \sup\{0,|1 - (1-K)|\}=K\le K$$
for all such $\epsilon,$ satisfying the conditions of the question.  But since 
$$|F-G(\epsilon)|_1 = \int_1^{1+\epsilon}|1 - (1-K)| \mathrm{d}x = K\epsilon,$$ 
as $\epsilon$ ranges from $0$ upwards the possible Kolmogorov distances range from $0$ upwards.  Thus the best possible bounds are $0$ and $\infty.$

Here's some intuition. 
First, the Kolmogorov metric is a probability whereas the Wasserstein metric is expressed in units of $x,$ so the two metrics aren't even comparable.  Since the latter can be arbitrarily large, we ought to suspect there is no upper bound.
Second, that thought suggests problems could arise by keeping the graphs of $F$ and $G$ within $K$ of each other (vertically) while somehow adjusting the area between them (the Wasserstein distance).  


*

*If we make $F$ and $G$ close everywhere except throughout a short interval where first one of them leaps upward and then, a little later (at higher values of $x$) the second leaps upward, this area can be made arbitrarily small. 

*But if we put a distance of $K$ between $F$ and $G$ over a wide range of $x,$ the area will be proportional to $K$ times that range, which can be made arbitrarily large.
The family of examples in this answer is one of the simplest that exhibits these behaviors, but it should be clear that many such families could be constructed.  Unless severe restrictions are imposed on $F$ and $G,$ then, the same result will hold.
