Improved lower bounds for embeddings into L₁

We simplify and improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into LI. In particular, we show that for infinitely many values of n there are n-point metric spaces of negative type that require a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a well-known SDP relaxation for SPARSEST-CUT. This result builds upon and improves the recent lower bound of (log log n)^1-6-o(1) due to Khot and Vishnoi [STOC 2005]. We also show that embedding the edit distance on {0, 1}ⁿ into L₁ requires a distortion of Ω(log n). This result simplifies and improves a very recent lower bound due to Khot and Naor [FOCS 2005].