The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric

We prove that the Cartesian product of octahedra $B_{1,\infty}^{n,m}=B_1^n\times \dots\times B_1^n$ ($m$ factors) is poorly approximated by spaces of half dimension in the mixed norm: $d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m})\ge cm$, $N=mn$. As a corollary, we find the order of linear widths of the Hölder–Nikolskii classes $H^r_p(\mathbb T^d)$ in the metric of $L_q$ in certain domains of variation of the parameters $(p,q)$.