I am trying to find the Lipschitz constant for the following function: $ f(\pi)=\left|\sum_{i=1}^{m}c_{\pi(i)}-\sum_{i=m+1}^{2m}c_{\pi(i)}\right|, $ where $c_i \in R$ and $\pi$ is a permutation of the set $\{1,..., 2m\}$ with counting metric $\rho_{2m}=\#\{i: \pi_1(i)\neq \pi_2(i); \pi_1, \pi_2 \in S_{2m}\}$.
I've got Lipschitz constant $2\lVert c\rVert_{\infty}$. I am wondering if one can do something better?