Concentration of mass and central limit properties of isotropic convex bodies

Author:
G. Paouris

Journal:
Proc. Amer. Math. Soc. **133** (2005), 565-575

MSC (2000):
Primary 52A20; Secondary 52A38, 52A40

DOI:
https://doi.org/10.1090/S0002-9939-04-07757-3

Published electronically:
September 20, 2004

MathSciNet review:
2093081

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the following question: Do there exist an absolute constant and a sequence tending to infinity with , such that for every isotropic convex body in and every the inequality holds true? Under the additional assumption that is 1-unconditional, Bobkov and Nazarov have proved that this is true with . The question is related to the central limit properties of isotropic convex bodies. Consider the spherical average . We prove that for every and every isotropic convex body in , the statements (A) ``for every , " and (B) ``for every , , where " are equivalent.

**1.**S. Alesker,*-estimate for the Euclidean norm on a convex body in isotropic position*, Geom. Aspects of Funct. Analysis (Lindenstrauss-Milman eds.), Oper. Theory Adv. Appl.**77**(1995), 1-4. MR**1353444 (96g:52004)****2.**M. Anttila, K.M. Ball and I. Perissinaki,*The central limit problem for convex bodies*, Trans. Amer. Math. Soc.**355**(2003), 4723-4735. MR**1997580****3.**K.M. Ball and I. Perissinaki,*The subindependence of coordinate slabs in balls*, Israel J. Math.**107**(1998), 289-299. MR**1658571 (99k:52012)****4.**S.G. Bobkov,*Remarks on the growth of -norms of polynomials*, Geom. Aspects of Funct. Analysis (Milman-Schechtman eds.), Lecture Notes in Math.**1745**(2000), 27-35. MR**1796711 (2002b:46016)****5.**J. Bourgain,*On the distribution of polynomials on high-dimensional convex sets*, Geom. Aspects of Funct. Analysis (Lindenstrauss-Milman eds.), Lecture Notes in Math.**1469**(1991), 127-137. MR**1122617 (92j:52007)****6.**S.G. Bobkov and A. Koldobsky,*On the central limit property of convex bodies*, Geom. Aspects of Funct. Analysis (Milman-Schechtman eds.), Lecture Notes in Math.**1807**(2003), 44-52.**7.**S.G. Bobkov and F.L. Nazarov,*On convex bodies and log-concave probability measures with unconditional basis*, Geom. Aspects of Funct. Analysis (Milman-Schechtman eds.), Lecture Notes in Math.**1807**(2003), 53-69.**8.**U. Brehm and J. Voigt,*Asymptotics of cross sections for convex bodies*, Beiträge Algebra Geom.**41**(2000), 437-454. MR**1801435 (2002i:52001)****9.**A. Carbery and J. Wright,*Distributional and -norm inequalities for polynomials over convex bodies in*, Math. Res. Lett.**8**(2001), 233-248. MR**1839474 (2002h:26033)****10.**R. Kannan, L. Lovasz and M. Simonovits,*Isoperimetric problems for convex bodies and a localization lemma*, Discrete Comput. Geom.**13**(1995), 541-559. MR**1318794 (96e:52018)****11.**V.D. Milman and A. Pajor,*Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed -dimensional space*, Geom. Aspects of Funct. Analysis (Lindenstrauss-Milman eds.), Lecture Notes in Math.**1376**(1989), 64-104. MR**1008717 (90g:52003)****12.**V.D. Milman and G. Schechtman,*Asymptotic Theory of Finite-Dimensional Normed Spaces*, Lecture Notes in Math.**1200**, Springer, Berlin (1986). MR**0856576 (87m:46038)****13.**G. Paouris,*-estimates for linear functionals on zonoids*, Geom. Aspects of Funct. Analysis (Milman-Schechtman eds.), Lecture Notes in Math.**1807**(2003), 211-222.**14.**R. Schneider,*Convex Bodies: The Brunn-Minkowski Theory*, Encyclopedia of Mathematics and its Applications**44**, Cambridge University Press, Cambridge (1993). MR**1216521 (94d:52007)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
52A20,
52A38,
52A40

Retrieve articles in all journals with MSC (2000): 52A20, 52A38, 52A40

Additional Information

**G. Paouris**

Affiliation:
Department of Mathematics, University of Crete, Iraklion 714-09, Greece

Email:
paouris@math.uoc.gr

DOI:
https://doi.org/10.1090/S0002-9939-04-07757-3

Keywords:
Isotropic convex bodies,
concentration of volume,
central limit theorem

Received by editor(s):
August 2, 2003

Published electronically:
September 20, 2004

Communicated by:
Nicole Tomczak-Jaegermann

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.