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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Concentration of $ 1$-Lipschitz maps into an infinite dimensional $ \ell^p$-ball with the $ \ell^q$-distance function


Author: Kei Funano
Journal: Proc. Amer. Math. Soc. 137 (2009), 2407-2417
MSC (2000): Primary 53C21, 53C23
Published electronically: March 12, 2009
MathSciNet review: 2495276
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Abstract: In this paper, we study the Lévy-Milman concentration phenomenon of $ 1$-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $ \ell^p$-ball with the $ \ell^q$-distance function for $ 1\leq p<q\leq +\infty$ is equivalent to the concentration to the real line.


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Additional Information

Kei Funano
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Email: sa4m23@math.tohoku.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09873-6
PII: S 0002-9939(09)09873-6
Keywords: mm-space, infinite dimensional $\ell ^p$-ball, concentration of $1$-Lipschitz maps, L\'evy group
Received by editor(s): August 25, 2008
Published electronically: March 12, 2009
Additional Notes: This work was partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
Communicated by: Mario Bonk
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.