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 L1. In particular, we show that for every n ≥ 1, there is an n-point metric space of negative type that requires a distortion of Ω(log log n) for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming 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 [Proc. of 46th FOCS, 2005]. We also show that embedding the edit distance metric on {0, 1}n into L1 requires a distortion of Ω(log n). This result simplifies and improves a very recent (log n)1/2−o(1) lower bound by Khot and Naor [Proc. of 46th FOCS, 2005].