  We show that any embedding of the level-k diamond graph of Newman and
Rabinovich into $L_p$, $1 < p \le 2$, requires distortion at least
$\sqrt{k(p-1) + 1}$. An immediate consequence is that there exist arbitrarily
large n-point sets $X \subseteq L_1$ such that any D-embedding of X into
$\ell_1^d$ requires $d \geq n^{\Omega(1/D^2)}$. This gives a simple proof of
the recent result of Brinkman and Charikar which settles the long standing
question of whether there is an $L_1$ analogue of the Johnson-Lindenstrauss
dimension reduction lemma.
