Kolmogorov widths of balls in mixed norms: the case of rigidity

We describe the set of parameters $(p_1,p_2,q_1,q_2)$ such that the balls $B_{q_1,q_2}^{s,b}$ are rigid in ${\mathrm{\ell }}_{q_1,q_2}^{s,b}$ metric i.e. they are poorly approximated by linear subspaces of dimension $\le (1-\epsilon )sb$, for large $s,b$. Thus we have settled an important qualitative case in the problem of estimating widths of balls in mixed norms. The proof combines lower bounds from our previous papers and a new construction for the approximation by linear subspaces in the so-called exceptional case.