Let $x,y\in\Bbb R ^N$ and $\pi ^x , \pi ^y \in \mathcal S_N$ such that $$x_{\pi^x (1)} \geq \ldots \geq x_{\pi^x (N)} ,\quad y_{\pi^y (1)} \geq \ldots \geq y_{\pi^y (N)}.$$
I think there should hold: $$\max_{k=1,\ldots,N} x_{\pi^x (k)} - y_{\pi^y(k)} \leq \max_{i=1,\ldots,N } x_i - y_i$$