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Abstract : |
The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky?s theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any >0, every n point metric space contains a subset of size at least n1? ? ? log(1/) which is embeddable in Hilbert space with O distortion. The bound on the distortion is tight up to the log(1/) factor. We further include a comprehensive study of various other aspects of this problem. Contents 1., |
