The product of octahedra is badly approximated in the ℓ2,1-metric

Abstract

We prove that the Cartesian product of octahedra B_1,∞^n,m = B₁ⁿ ×···× B₁ⁿ (m factors) is poorly approximated by spaces of half dimension in the mixed norm: d_N/2(B_1,∞^n,m, ℓ_2,1^n,m) ≥ cm, N = mn. As a corollary, we find the order of linear widths of the Hölder–Nikol’skii classes H_p^r(T^d) in the metric of L_q in certain domains of variation of the parameters (p, q).