Kolmogorov widths of balls in mixed norms: the case of rigidity
Yuri Malykhin, Konstantin Ryutin
TL;DR
The paper addresses the problem of describing rigidity for finite-dimensional balls $B_{p_1,p_2}^{s,b}$ in the mixed-norm space $\ell_{q_1,q_2}^{s,b}$. The main approach combines lower bounds from prior work (MR25 and MR17) with a new partition-based construction to handle the exceptional regime. The main result yields a complete description: rigidity holds under the inequalities (i) and (ii) away from the exceptional case; otherwise, upper bounds with decay $d_n \le t^{-\gamma} d_0$ are established via an operator $D_\Gamma$ built from an $(m,r,l)$-partition. These findings advance understanding of widths for mixed-norm balls and introduce novel combinatorial tools for block-structured approximation in high dimensions.
Abstract
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 $\ell_{q_1,q_2}^{s,b}$ metric i.e. they are poorly approximated by linear subspaces of dimension $\le (1-\varepsilon)sb$, for large $s, b$. 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.