Comptes Rendus
no. 3
Functional Analysis
Concentration of mass on isotropic convex bodies
[Concentration de masse pour les corps convexes isotropes]

Présenté par : Gilles Pisier

Grigoris Paouris1
1 Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece
Comptes Rendus. Mathématique, Volume 342 (2006) no. 3, pp. 179-182.

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

Prob({x∈K:‖x‖2⩾cnLKt})⩽exp(−nt)
pour tout t⩾1, où LK désigne la constante d'isotropie.

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({x∈K:‖x‖2⩾cnLKt})⩽exp(−nt)
for every t⩾1, where LK denotes the isotropic constant.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.11.018
Grigoris Paouris 1

1 Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece 
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Grigoris Paouris. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.018/

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

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

[3] S.G. Bobkov; F.L. Nazarov 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] J. Bourgain 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] A. Litvak; V.D. Milman; G. Schechtman Averages of norms and quasi-norms, Math. Ann., Volume 312 (1998), pp. 95-124

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

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

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

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

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

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

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

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