Kolmogorov widths of an intersection of anisotropic finite-dimensional balls in $l_q^k$ for $1\le q\le 2$

In this paper, order estimates for the Kolmogorov $n$-widths of an intersection of an arbitrary family of balls ${\nu }_{\alpha }{B}_{{\bar{p}}_{\alpha }}^{\bar{k}}$ in $l_q^k$ are obtained for $1\le q\le 2$, $n\le \frac{k}{2}$. Here ${\bar{p}}_{\alpha }=(p_{\alpha ,1},\dots ,p_{\alpha ,d})$, $\bar{k}=(k_1,\dots ,k_d)$, $k=k_1\dots k_d$, $B_{{\bar{p}}_{\alpha }}^{\bar{k}}$ is the unit ball with respect to the anisotropic norm given by the vector ${\bar{p}}_{\alpha }$.