This paper concerns the problem of the Kolmogorov $n$-width, the linear
$n$-width, the Gel'fand $n$-width and the Bernstein $n$-width of Sobolev
classes of the periodic multivariate functions in the space $L_{\bf p}
(T^d)$ and the average Bernstein $\sz$-width, average Kolmogorov $\sz$-widths,
the average linear $\sz$-widths of Sobolev classes of the multivariate
functions in the space $L_{\bf p}(R^d)
$, where ${\bf p}=(p_1,\cdots,p_d),1\le p_j<\infty,
j=1,2,\cdots,d,$ or $p_j=\infty, j=1,2,\cdots,d$.
Their weak asymptotic behaviors are established for
the corresponding quantities.
