On a Kantorovich-Rubinstein inequality

An easy consequence of Kantorovich-Rubinstein duality is the following: if f : [ 0 , 1 ] d → R Lipschitz and { x … N } ⊂ then | ∫ ( ) − ∑ k = ≤ ‖ ∇ L ∞ ⋅ W δ where denotes 1−Wasserstein (or Earth Mover's) Distance. We prove a similar inequality with smaller norm on larger Wasserstein distance instead ). Our sharp when points are very regular, i.e. ∼ / . This prompts question whether these two inequalities specific instances an entire underlying family estimates capturing between transport function space.