On the best uniform polynomial approximation to the checkmark function

The best uniform polynomial approximation of the checkmark function f(x)=|x−α| is considered, as α varies in (−1,1). For each fixed degree n, minimax error En(α) shown to be piecewise analytic α. In addition, feature n−1 linear decreasing/increasing sections, called V-shapes. points alternation set are proven and monotone increasing their dynamics completely characterized. We also prove a conjecture Shekhtman that for odd has local maximum at α=0.