On a Kantorovich-Rubinstein Inequality

Abstract An easy consequence of Kantorovich-Rubinstein duality is the following: if f : [ 0 , 1 ] d → R is Lipschitz and { x 1 , … , x N } ⊂ [ 0 , 1 ] d , then | ∫ [ 0 , 1 ] d f ( x ) d x − 1 N ∑ k = 1 N f ( x k ) | ≤ ‖ ∇ f ‖ L ∞ ⋅ W 1 ( 1 N ∑ k = 1 N δ x k , d x ) , where W 1 denotes the 1−Wasserstein (or Earth Mover's) Distance. We prove a similar inequality with a smaller norm on ∇f and a larger Wasserstein distance ( W ∞ instead of W 1 ). Our inequality is sharp when the points are very regular, i.e. W ∞ ∼ 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.