Kolmogorov Widths of the Besov Classes $B^1_{1,\theta }$ and Products of Octahedra

We find the decay orders of the Kolmogorov widths of some Besov classes related to $W^1_1$ (the behavior of the widths for the class $W^1_1$ remains unknown): $d_n(B^1_{1,\theta }[0,1],L_q[0,1])\asymp n^{-1/2}\log ^{\max \{1/2,1-1/\theta \}}n$ for $2<q<\infty$ and $1\le \theta \le \infty$. The proof relies on the lower bound for the width of a product of octahedra in a special norm (maximum of two weighted $\ell _{q_i}$ norms). This bound generalizes B. S. Kashin's theorem on the widths of octahedra in $\ell _q$.