  We prove that Hilbert space is distortable and, in fact, arbitrarily
distortable. This means that for all lambda >1 there exists an equivalent norm
|.| on l_2 such that for all infinite dimensional subspaces Y of l_2 there
exist x,y in Y with ||x||_2 = ||y||_2 =1 yet |x| >lambda |y|.
  We also prove that if X is any infinite dimensional Banach space with an
unconditional basis then the unit sphere of X and the unit sphere of l_1 are
uniformly homeomorphic if and only if X does not contain l_infty^n's uniformly.
