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We show that any embedding of the level k diamond graph of Newman and Rabi- novich (6) into Lp, 1 < p • 2, requires distortion at least p k(p ¡ 1) + 1. An imme- diate corollary is that there exist arbitrarily large n-point sets X µ L1 such that any D-embedding of X into 'd1 requires d ‚ n›(1=D 2 ) . This gives a simple proof of a recent result of Brinkman and Charikar (2) which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma (4).

Embedding the diamond graph in Lp and dimension reduction in L1