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We exhibit an infinite doubling metric space (X,d) such that for any non-expansive f : X - L1, there exists a pair x,y ∈ X with d(x,y) arbitrarily large, and such that |f(x)-f(y)\|1/d(x,y) ≲ √log log d(x,y)}/(log d(x,y)). As a consequence, we show that there are n-point doubling metrics which require distortion Ω(√{log n/(log log n)}) into L1, matching the upper bound of [Gupta-Krauthgamer-Lee, FOCS'03] up to a factor of O(√log log n). The best previous lower bound for doubling spaces, due to [Cheeger-Kleiner-Naor, FOCS'09] was of the form (log n)δ for some small, unspecified value of δ 0. Furthermore, this gives a nearly optimal integrality gap for a weak version of the SDP for the general Sparsest Cut Problem. The weak SDP suffices for all known rounding algorithms, and the best previous gap was of the order (log n)1/4/(log log n) [Lee-Moharrami, STOC'10].

Near-optimal distortion bounds for embedding doubling spaces into L 1

Near-optimal distortion bounds for embedding doubling spaces into L ₁