  We give an explicit (in particular, deterministic polynomial time)
construction of subspaces X of R^N of dimension (1-o(1))N such that for every
element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log
N)^{log log log N}. If we are allowed to use N^{o(1)} random bits, this factor
can be improved to poly(log N).
  Our construction makes use of unbalanced bipartite graphs to impose local
linear constraints on vectors in the subspace, and our analysis relies on
expansion properties of the graph. This is inspired by similar constructions of
error-correcting codes.
