Geodesic
Properties of random sections of an $N$-dimensional cube
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1983), pp. 8-11 
B. S. Kashin. Properties of random sections of an $N$-dimensional cube. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1983), pp. 8-11. http://geodesic.mathdoc.fr/item/VMUMM_1983_3_a1/
@article{VMUMM_1983_3_a1,
     author = {B. S. Kashin},
     title = {Properties of random sections of an $N$-dimensional cube},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {8--11},
     year = {1983},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1983_3_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $R^N$.