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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The surface measure and cone measure on the sphere of $ \ell_p^n$


Author: Assaf Naor
Journal: Trans. Amer. Math. Soc. 359 (2007), 1045-1079
MSC (2000): Primary 52A20, 60B11
Published electronically: September 11, 2006
MathSciNet review: 2262841
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Abstract: We prove a concentration inequality for the $ \ell_q^n$ norm on the $ \ell_p^n$ sphere for $ p,q>0$. This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance between the cone measure and surface measure on the sphere of $ \ell_p^n$. In particular, we obtain a significant strengthening of the inequality derived by Naor and Romik (2003), and calculate the precise dependence of the constants that appeared there on $ p$.


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Additional Information

Assaf Naor
Affiliation: Department of Mathematics, Hebrew University, Givaat-Ram, Jerusalem, Israel
Address at time of publication: Microsoft Research, One Microsoft Way, Redmond, Washington 98052-6399
Email: anaor@microsoft.com

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03939-0
PII: S 0002-9947(06)03939-0
Keywords: Geometry of $\ell_p^n$, cone measure, surface measure, concentration inequalities, convex geometry
Received by editor(s): May 14, 2001
Received by editor(s) in revised form: November 22, 2004
Published electronically: September 11, 2006
Additional Notes: This work was partially supported by BSF and the Clore Foundation, and is part of the author’s Ph.D. thesis prepared under the supervision of Professor Joram Lindenstrauss.
Article copyright: © Copyright 2006 American Mathematical Society