  In this paper we find the orders of decay for Kolmogorov widths of some Besov
classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains
unknown): $$ d_n(B^1_{1,\theta}[0,1],L_q[0,1])\asymp
n^{-1/2}\log^{\max(\frac12,1-\frac{1}{\theta})}n,\quad 2<q<\infty. $$ The proof
relies on the lower bound for widths of product of octahedra in a special norm
(maximum of two weighted $\ell_q$ norms). This bound generalizes the theorem of
B.S.~Kashin on widths of octahedra in $\ell_q^N$.
