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Some applications of Ball's extension theorem

Authors: Manor Mendel and Assaf Naor
Journal: Proc. Amer. Math. Soc. 134 (2006), 2577-2584
MSC (2000): Primary 46B20; Secondary 51F99
Published electronically: February 17, 2006
MathSciNet review: 2213735
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Abstract: We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type $ 2$ problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice $ \{0,1,\ldots,m\}^n$, equipped with the $ \ell_p^n$ metric, in any $ 2$-uniformly convex Banach space is of order $ \min \left\{n^{\frac12-\frac{1}{p}},m^{1-\frac{2}{p}}\right\}$.

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Manor Mendel
Affiliation: Department of Computer Science, California Institute of Technology, Pasadena, California 91125
Address at time of publication: Computer Science Division, The Open University of Israel, 108 Ravutski Street, P.O.B. 808, Raanana 43107, Israel

Assaf Naor
Affiliation: Theory Group, Microsoft Research, Redmond, Washington 90852

Keywords: Lipschitz extension, bi-Lipschitz embeddings
Received by editor(s): January 27, 2005
Received by editor(s) in revised form: March 18, 2005
Published electronically: February 17, 2006
Communicated by: David Preiss
Article copyright: © Copyright 2006 American Mathematical Society

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