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 the $$\ell_{q_1,q_2}^{s,b}$$ metric; i.e., they are poorly approximated by linear subspaces of dimension $${\le}\,(1-\varepsilon)sb$$ for large $$s$$ and $$b$$ . Thus we have settled an important qualitative case in the problem of estimating the 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.