Functional Analysis
Concentration of mass on isotropic convex bodies
Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 179-182.

We establish sharp concentration of mass for isotropic convex bodies: there exists an absolute constant c>0 such that if K is an isotropic convex body in Rn, then

Prob({xK:x2cnLKt})exp(nt)
for every t1, where LK denotes the isotropic constant.

Nous démontrons qu'il existe une constante absolue c>0, telle que, si K est un corps convexe isotrope, alors

Prob({xK:x2cnLKt})exp(nt)
pour tout t1, où LK désigne la constante d'isotropie.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.11.018
Paouris, Grigoris 1

1 Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece
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Paouris, Grigoris. Concentration of mass on isotropic convex bodies. Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 179-182. doi : 10.1016/j.crma.2005.11.018. http://www.numdam.org/articles/10.1016/j.crma.2005.11.018/

[1] Alesker, S. ψ2-estimate for the Euclidean norm on a convex body in isotropic position (Lindenstrauss, X.; Milman, V.D., eds.), Geom. Aspects of Funct. Analysis, Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 1-4

[2] Bobkov, S.G.; Nazarov, F.L. On convex bodies and log-concave probability measures with unconditional basis (Milman, V.D.; Schechtman, G., eds.), Geom. Aspects of Funct. Analysis, Lecture Notes in Math., vol. 1807, Springer, 2003, pp. 53-69

[3] Bobkov, S.G.; Nazarov, F.L. Large deviations of typical linear functionals on a convex body with unconditional basis, Stochastic Inequalities and Applications, Progr. Probab., vol. 56, Birkhäuser, Basel, 2003, pp. 3-13

[4] Bourgain, J. Random points in isotropic convex bodies, Convex Geometric Analysis, Berkeley, CA, 1996, Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 53-58

[5] O. Guédon and G. Paouris, Concentration of mass on the Schatten classes, Preprint

[6] Litvak, A.; Milman, V.D.; Schechtman, G. Averages of norms and quasi-norms, Math. Ann., Volume 312 (1998), pp. 95-124

[7] Lutwak, E.; Yang, D.; Zhang, G. Lp affine isoperimetric inequalities, J. Differential Geom., Volume 56 (2000), pp. 111-132

[8] Lutwak, E.; Zhang, G. Blaschke–Santaló inequalities, J. Differential Geom., Volume 47 (1997), pp. 1-16

[9] Milman, V.D.; Pajor, A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space (Lindenstrauss, X.; Milman, V.D., eds.), Geom. Aspects of Funct. Analysis, Lecture Notes in Math., vol. 1376, Springer, 1989, pp. 64-104

[10] Milman, V.D.; Schechtman, G. Global versus local asymptotic theories of finite-dimensional normed spaces, Duke Math. J., Volume 90 (1997), pp. 73-93

[11] Paouris, G. Concentration of mass and central limit properties of isotropic convex bodies, Proc. Amer. Math. Soc., Volume 133 (2005) no. 2, pp. 565-575

[12] Paouris, G. On the Ψ2-behavior of linear functionals on isotropic convex bodies, Studia Math., Volume 168 (2005) no. 3, pp. 285-299

[13] Rudelson, M. Random vectors in the isotropic position, J. Funct. Anal., Volume 164 (1999), pp. 60-72

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