Abstract
It is shown that anyn point metric space is up to logn lipeomorphic to a subset of Hilbert space. We also exhibit an example of ann point metric space which cannot be embedded in Hilbert space with distortion less than (logn)/(log logn), showing that the positive result is essentially best possible. The methods used are of probabilistic nature. For instance, to construct our example, we make use of random graphs.
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Bourgain, J. On lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math. 52, 46–52 (1985). https://doi.org/10.1007/BF02776078
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DOI: https://doi.org/10.1007/BF02776078