Widths and rigidity of unconditional sets and random vectors

We prove that any unconditional set in $\mathbb R^N$ that is invariant under cyclic shifts of coordinates is rigid in $\ell_q^N$, $1\leqslant q\leqslant 2$, i.e. it can not be well approximated by linear spaces of dimension essentially smaller than $N$. We apply the approach of E. D. Gluskin to the setting of averaged Kolmogorov widths of unconditional random vectors or vectors of independent mean zero random variables, and prove their rigidity. These results are obtained using a general lower bound for the averaged Kolmogorov width via weak moments of biorthogonal random vector. This paper continues the study of the rigidity initiated by the first author. We also provide several corollaries including new bounds for Kolmogorov widths of mixed norm balls.