Embedding the diamond graph in $L_p$ and dimension reduction in $L_1$

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.