TY - JOUR

T1 - Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem

AU - Naor, Assaf

AU - Tao, Terence

N1 - Funding Information:
Acknowledgements. We thank Manor Mendel for helpful discussions on the Thorup–Zwick lower bound. A. N. is supported by NSF grants CCF-0635078 and CCF-0832795, BSF grant 2006009, and the Packard Foundation. T. T. is supported by a grant from the MacArthur foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.

PY - 2012/12

Y1 - 2012/12

N2 - We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is D-equivalent to an ultrametric, where D ∈ (2,∞) is a prescribed target distortion. Since this procedure works for D arbitrarily close to the nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much simpler probabilistic proof of the main result of [3], answering a question from [12], and yielding the best known bounds in the nonlinear Dvoretzky theorem. Our method utilizes a sequence of random scales at which a given metric space is fragmented. As in many previous randomized arguments in embedding theory, these scales are chosen irrespective of the geometry of the metric space in question. We show that our bounds are sharp if one utilizes such a "scale-oblivious" fragmentation procedure.

AB - We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is D-equivalent to an ultrametric, where D ∈ (2,∞) is a prescribed target distortion. Since this procedure works for D arbitrarily close to the nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much simpler probabilistic proof of the main result of [3], answering a question from [12], and yielding the best known bounds in the nonlinear Dvoretzky theorem. Our method utilizes a sequence of random scales at which a given metric space is fragmented. As in many previous randomized arguments in embedding theory, these scales are chosen irrespective of the geometry of the metric space in question. We show that our bounds are sharp if one utilizes such a "scale-oblivious" fragmentation procedure.

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U2 - 10.1007/s11856-012-0039-7

DO - 10.1007/s11856-012-0039-7

M3 - Article

AN - SCOPUS:84871740206

VL - 192

SP - 489

EP - 504

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -