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In this paper we find the orders of decay for Kolmogorov widths of some Besovclasses related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remainsunknown): $$ d_n(B^1_{1,\theta}[0,1],L_q[0,1])\asympn^{-1/2}\log^{\max(\frac12,1-\frac{1}{\theta})}n,\quad 2<q<\infty. $$ The proofrelies 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 ofB.S.~Kashin on widths of octahedra in $\ell_q^N$.

Kolmogorov widths of Besov classes $B^1_{1,\theta}$ and products of octahedra