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We prove that the cartesian product of octahedra $B_{1,\infty}^{n,m}=B_1^n\times\ldots\times B_1^n$ ($m$ octahedra) is badly approximated by half--dimensional subspaces in mixed--norm: $d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m})\ge cm$, $N=mn$.
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