It is well known that $R^N$ has subspaces of dimension proportional to $N$ on which the $ell_1$ norm is equivalent to the $ell_2$ norm; however, no explicit constructions are known. Extending earlier work by Artstein--Avidan and Milman, we prove that such a subspace can be generated using $O(N)$ random bits.

It is well known that $\R^N$ has subspaces of dimension
proportional to $N$ on which the $\ell_1$ norm is equivalent to the
$\ell_2$ norm; however, no explicit constructions are known.
Extending earlier work by Artstein--Avidan and Milman, we prove that
such a subspace can be generated using $O(N)$ random bits.