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 ε > 0, every n point metric space contains a subset of size at least n1-ε which is embeddable in Hilbert space with O ( log(1/ε)/ε) distortion. The bound on the distortion is tight up to the log(1/ε) factor. We further include a comprehensive study of various other aspects of this problem.
Original language | English (US) |
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Pages (from-to) | 643-709 |
Number of pages | 67 |
Journal | Annals of Mathematics |
Volume | 162 |
Issue number | 2 |
DOIs | |
State | Published - Sep 2005 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)