A Complete Description of the Relative Widths of Sobolev Classes in the Uniform Metric

We consider the width of the Sobolev class of $2\pi$-periodic functions with ${\|f^{(r)}\|_\infty\le 1}$ with respect to the set of functions $g$ such that $\|g^{(r)}\|_\infty\le M$ in the uniform metric $K_n:=K_n(W^r_\infty,MW^r_\infty,L_\infty)$. We prove a lower bound on $K_n$ for $M=1+\varepsilon$ with small $\varepsilon$. This bound together with earlier results completes the analysis of the behavior of $K_n$.