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

Abstract

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): $$d_n(B_{1,\theta }^1[0,1],L_q[0,1])\asymp n^{-1/2}{\log }^{\max (\frac{1}{2},1-\frac{1}{\theta })}n,2<q<\infty .$$ 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 ${\ell }_q^N$.