Kolmogorov \(n\)-width of some finite-dimensional sets in a mixed norm
Denote by \(V^n_k\) the convex hull in \(\mathbb{R}^n\) of the points having \(k\) coordinates equal to \(\pm 1\) and the remaining equal to zero, and denote \(V_{k,\infty}^{n,m} =V^k_n\times \cdots \times V_n^k\). For \(x\in\mathbb{R}^{nm}\), let \[ |x|_{l^{n,m}_{p,q}} = \left(\sum^m_{s=1} \Bigl(\sum_{k\in\Delta_s} |x_k|^p \Bigr)^{q/p} \right)^{1/q} \quad \text{with} \quad \Delta_s= \bigl\{k\in N;(s-1) n<k<sn\bigr\}. \] The following asymptotics for the Kolmogorov width \(d_N(V^{n,m}_{k,\infty}, l^{n,m}_{p,q})=d\) are determined: \(d\asymp k^{1/p} m^{1/q}\) provided \(1<q \leq\infty\), \(N<nm/2\), \(1\leq k\leq n\), \(p=2\) or \(1<p\leq \min(q,2)\).
scientific article; zbMATH DE number 898918