Properties of random sections of an $N$-dimensional cube

An answer is given to a question of T. Figiel and W. Johnson. It is shown that $0<\alpha<1$, $1\le n\le N^\alpha$,
$$ \int d(l^N_\infty\cap L,l^n_2)\,d\mu_{N,n}\le C_\alpha\max(n^{1/2}\ln^{-1/2}N,1), $$
where $d(X,Y)$ is the Banach–Mazur distance between normed spaces $X$ and $Y$, $L$ are $n$-dimensional subspaces in $R^N$, and $\mu_{N,n}$ is the invariant measure on the set of all $n$-dimensional subspaces in $R^N$.