On a Kantorovich-Rubinstein inequality

Abstract: An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left{x_1, \dots, x_N \right} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N}{f(x_k)} \right| \leq \left| \nabla f \right|{L^{\infty}} \cdot W_1\left( \frac{1}{N} \sum{k=1}^{N}{\delta_{x_k}} , dx\right),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Mover's) Distance. We prove another such inequality with a smaller norm on $\nabla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e. $W_{\infty} \sim N^{-1/d}$. This prompts the question whether these two inequalities are specific instances of an entire underlying family of estimates capturing a duality between transport distance and function space.