## The surface measure and cone measure on the sphere of $\ell _p^n$

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- by Assaf Naor PDF
- Trans. Amer. Math. Soc.
**359**(2007), 1045-1079 Request permission

## 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

- Emil Artin,
*The gamma function*, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York-Toronto-London, 1964. Translated by Michael Butler. MR**0165148** - Milla Anttila, Keith Ball, and Irini Perissinaki,
*The central limit problem for convex bodies*, Trans. Amer. Math. Soc.**355**(2003), no. 12, 4723–4735. MR**1997580**, DOI 10.1090/S0002-9947-03-03085-X - Juan Arias-de-Reyna, Keith Ball, and Rafael Villa,
*Concentration of the distance in finite-dimensional normed spaces*, Mathematika**45**(1998), no. 2, 245–252. MR**1695717**, DOI 10.1112/S0025579300014182 - J. Arias-de-Reyna and R. Villa,
*The uniform concentration of measure phenomenon in $l^n_p$ $(1\leq p\leq 2)$*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 13–18. MR**1796708**, DOI 10.1007/BFb0107203 - F. Barthe, M. Csörnyei, and A. Naor,
*A note on simultaneous polar and Cartesian decomposition*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 1–19. MR**2083383**, DOI 10.1007/978-3-540-36428-3_{1} - Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor,
*A probabilistic approach to the geometry of the $l^n_p$-ball*, Ann. Probab.**33**(2005), no. 2, 480–513. MR**2123199**, DOI 10.1214/009117904000000874 - F. Barthe and A. Naor,
*Hyperplane projections of the unit ball of $l^n_p$*, Discrete Comput. Geom.**27**(2002), no. 2, 215–226. MR**1880938**, DOI 10.1007/s00454-001-0066-3 - Keith Ball and Irini Perissinaki,
*The subindependence of coordinate slabs in $l^n_p$ balls*, Israel J. Math.**107**(1998), 289–299. MR**1658571**, DOI 10.1007/BF02764013 - Richard Durrett,
*Probability: theory and examples*, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR**1609153** - Persi Diaconis and David Freedman,
*A dozen de Finetti-style results in search of a theory*, Ann. Inst. H. Poincaré Probab. Statist.**23**(1987), no. 2, suppl., 397–423 (English, with French summary). MR**898502** - M. Gromov and V. D. Milman,
*Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces*, Compositio Math.**62**(1987), no. 3, 263–282. MR**901393** - RafałLatała,
*Estimation of moments of sums of independent real random variables*, Ann. Probab.**25**(1997), no. 3, 1502–1513. MR**1457628**, DOI 10.1214/aop/1024404522 - A. A. Mogul′skiĭ,
*de Finetti-type results for $l_p$*, Sibirsk. Mat. Zh.**32**(1991), no. 4, 88–95, 228 (Russian); English transl., Siberian Math. J.**32**(1991), no. 4, 609–616 (1992). MR**1142071**, DOI 10.1007/BF00972979 - Assaf Naor and Dan Romik,
*Projecting the surface measure of the sphere of $\scr l_p^n$*, Ann. Inst. H. Poincaré Probab. Statist.**39**(2003), no. 2, 241–261 (English, with English and French summaries). MR**1962135**, DOI 10.1016/S0246-0203(02)00008-0 - 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**, DOI 10.1214/aop/1176990346 - J. Arias-de-Reyna and R. Villa,
*The uniform concentration of measure phenomenon in $l^n_p$ $(1\leq p\leq 2)$*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 13–18. MR**1796708**, DOI 10.1007/BFb0107203 - G. Schechtman and J. Zinn,
*On the volume of the intersection of two $L^n_p$ balls*, Proc. Amer. Math. Soc.**110**(1990), no. 1, 217–224. MR**1015684**, DOI 10.1090/S0002-9939-1990-1015684-0 - G. Schechtman and J. Zinn,
*Concentration on the $l^n_p$ ball*, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 245–256. MR**1796723**, DOI 10.1007/BFb0107218 - Michael Schmuckenschläger,
*CLT and the volume of intersections of $l^n_p$-balls*, Geom. Dedicata**85**(2001), no. 1-3, 189–195. MR**1845607**, DOI 10.1023/A:1010353121014

## 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
- 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.
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**359**(2007), 1045-1079 - MSC (2000): Primary 52A20, 60B11
- DOI: https://doi.org/10.1090/S0002-9947-06-03939-0
- MathSciNet review: 2262841