Shape-preserving linear n-width of unit balls in $C[0,1]$

Let $D^k$, $k$ is a natural number or zero, be the $k$-th differential operator, defined in $C^k(X)$, $X=[0,1]$, and let $C$ be a cone in $C^k(X)$. Let us denote $\delta_n^k(A,C)_{C(X)}:=\inf_{L_n(C)\subset C}\sup_{f\in A}\|D^kf-D^kL_nf\|_{C(X)}$ linear relative $n$-width of set $A\subset C^k(X)$ in $C(X)$ for $D^k$ with constraint $C$. In this paper we estimate linear relative $n$-width of some balls in $C(X)$ for $D^k$ with constraint $C=\{f\in C^k(X):D^kf\ge0\}$.