Box-constrained monotone $L_\infty$-approximations and Lipschitz-continuous regularized functions

Let $f:[0,1]\to[0,1]$ be a nondecreasing function. The main goal of this work is to provide a regularized version, say $\tilde f_L$, of $f$. Our choice will be a best $L_\infty$-approximation to $f$ in the set of functions $h:[0,1]\to[0,1]$ which are Lipschitz-continuous, for a fixed Lipschitz norm bound $L$, and verify the boundary restrictions $h(0)=0$ and $h(1)=1$. Our findings allow to characterize a solution through a monotone best $L_\infty$-approximation to the Lipschitz regularization of $f$. This is seen to be equivalent to follow the alternative way of the average of the Pasch-Hausdorff envelopes. We include results showing stability of the procedure as well as directional differentiability of the $L_\infty$-distance to the regularized version. This problem is motivated within a statistical problem involving trimmed versions of distribution functions as to measure the level of contamination discrepancy from a fixed model.