Kolmogorov Widths of Balls in Mixed Norms: The Case of Rigidity

We describe the set of parameters (p₁,p₂,q₁,q₂) such that the balls Bq₁,q₂s,b are rigid in the q₁,q₂s,b metric; i.e., they are poorly approximated by linear subspaces of dimension ≤\,(1-ε)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.