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

Author(s): Assaf Naor
Journal: Trans. Amer. Math. Soc. 359 (2007), 1045-1079.
MSC (2000): Primary 52A20, 60B11
Posted: September 11, 2006
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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$.

References:

[A]
E. Artin, The Gamma Function, Holt, Rinehart and Winston, 1964. MR 0165148 (29:2437)

[ABP]
M. Antilla, K. Ball and I. Perissinaki, The central limit problem for convex bodies, Trans. Amer. Math. Soc. 355 (2003), no. 12, 4723-4735.MR 1997580 (2005b:52010)

[A-d-RBV]
J. Arias-de-Reyna, K. Ball and R. Villa, Concentration of the distance in finite-dimensional normed spaces, Mathematika 45 (1998), no. 2, 245-252. MR 1695717 (2000b:46013)

[A-d-RV]
J. Arias-de-Reyna and R. Villa, The uniform concentration of measure phenomenon in $ \ell_p^n (1\le p\le 2)$. Geometric aspects of functional analysis, Lecture Notes in Math. 1745, 13-18, Springer, Berlin, 2000. MR 1796708 (2002b:46013a)

[BCN]
F. Barthe, M. Csörnyei and A. Naor, A note on simultaneous polar and Cartesian decomposition, Geometric aspects of functional analysis, Lecture Notes in Math. 1807, 1-19, Springer, Berlin, 2003.MR 2083383 (2005k:28011)

[BGMN]
F. Barthe, O. Guedon, S. Mendelson and A. Naor, A probabilistic approach to the geometry of the $ \ell_p^n$ ball, Ann. Probab. 33 (2005), no. 2, 480-513. MR 2123199

[BN]
F. Barthe and A. Naor, Hyperplane projections of the unit ball of $ \ell_p^n$, Discrete Comput. Geom. 27 (2002), no. 2, 215-226.MR 1880938 (2002k:52005)

[BP]
K. Ball and I. Perissinaki, The subindependence of coordinate slabs in $ \ell_p^n$ balls, Israel J. Math. 107 (1998), 289-299.MR 1658571 (99k:52012)

[D]
R. Durrett, Probability theory and examples, Duxbury Press, 1996. MR 1609153 (98m:60001)

[DF]
P. Diaconis and D. Freedman, A dozen de Finetti-style results in search of a theory. Ann. Inst. Henri Poincare 23, 397-423.MR 0898502 (88f:60072)

[GM]
M. Gromov and V.D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Composito Math. 62 (1987), no. 3, 263-282. MR 0901393 (89f:46031)

[L]
R. \Latala, Estimation of moments of sums of independent real random variables, Ann. Probab. 25 (1997), no. 3, 1502-1513.MR 1457628 (98h:60021)

[M]
A.A. Mogul'skii, de Finetti-type results for $ \ell_p$. (Russian), Sibirsk. Mat. Zh. 32 (1991), no. 4, 88-95. Translation in Siberian Math. J. 32 (1992), no. 4, 609-616. MR 1142071 (93b:60029)

[NR]
A. Naor and D. Romik, Projecting the surface measure of the sphere of $ \ell_p^n$, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 2, 241-261. MR 1962135 (2004d:60063)

[RaR]
S.T. Rachev and L. Rüschendorf, Approximate independence of distributions on spheres and their stability properties, Ann. Probab. 19 (1991), no. 3, 1311-1337. MR 1112418 (92k:60021)

[Sche]
G. Schechtman, An editorial comment on the preceding paper: ``The uniform concentration of measure phenomenon in $ \ell_p^n$ ( $ 1\le p\le 2$)''. Geometric aspects of functional analysis, 19-20. Lecture Notes in Math., 1745, Springer, Berlin, 2000.MR 1796709 (2002b:46013b)

[SZ1]
G. Schechtman and J. Zinn, On the volume of the intersection of two $ L_p^n$ balls, Proc. Amer. Math. Soc. 110 (1990), 1, 217-224.MR 1015684 (91c:46027)

[SZ2]
G. Schechtman and J. Zinn, Concentration on the $ \ell_p^n$ Ball, Geometric aspects of functional analysis, 245-256, Lecture Notes in Math., 1745, Springer, Berlin, 2000.MR 1796723 (2002b:46014)

[Schm]
M. Schmuckenschläger, C.L.T. and the volume of the intersection of two $ L_p^n$ balls, Geom. Dedicata 85 (2001), no. 1-3, 189-195.MR 1845607 (2002e:46013)

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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: 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
Posted: 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.
Copyright of article: Copyright 2006, American Mathematical Society

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