Kolmogorov Widths of an Intersection of Anisotropic Finite-Dimensional Balls in $$l^k_q$$ for $$1\le q \le 2$$

Order estimates for the Kolmogorov $$n$$ -widths of the intersection of any family of balls $$\nu_\alpha B^{\overline{k}}_{\overline{p}_\alpha}$$ in the space $$l_q^k$$ for $$1\le q\le 2$$ and $$n\le k/2$$ are obtained; here $$\overline{p}_\alpha=(p_{\alpha,1}, \dots, p_{\alpha,d})$$ , $$\overline{k}=(k_1, \dots, k_d)$$ , $$k=k_1\dots k_d$$ , and $$B^{\overline{k}}_{\overline{p}_\alpha}$$ is the unit ball with respect to the anisotropic norm determined by the vector $$\overline{p}_\alpha$$ .