Embedding the diamond graph in Lp and dimension reduction in L1

AbstractWe show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into Lp, 1 < p ≤ 2, requires distortion at least $$ \sqrt{k(p-1) + 1} $$. An immediate corollary is that there exist arbitrarily large n-point sets $$ X \subseteq L_1 $$ such that any D-embedding of X into $$ \ell^d_1 $$ requires $$ d \geq n^{\Omega(1/D^2)} $$. This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].