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 $$ left| int_{[0,1]^d} f(x) dx - frac{1}{N} sum_{k=1}^{N}{f(x_k)} right| leq left| abla f right|_{L^{infty}} cdot W_1left( frac{1}{N} sum_{k=1}^{N}{delta_{x_k}} , dxright),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Movers) Distance. We prove another such inequality with a smaller norm on $ abla 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.
Download