Comptes Rendus
no. 10
Randomizing properties of convex high-dimensional bodies and some geometric inequalities
[Les propriétés aléatoires des corps convexes de grande dimension et une inégalité géométrique]
Efim Gluskin1 ; Vitali Milman1
1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 875-879.

Nous étudions les propriétés des corps convexes liées à la distribution uniforme. En particulier, nous démontrons une borne inférieure pour la norme d'une somme de vecteurs aléatoires distribués géometriquement. Cette borne généralise le cas des vecteurs ayant la même distribution déjà étudié par Bourgain, Meyer, Milman et Pajor, et résout un problème posé par eux. Un autre corollaire énonce que chaque espace normé de dimension finie est de « cotype 2 aléatoire ».

Properties of convex bodies related to uniform distribution are studied. In particular, a low bound for the norm of the sum of independent geometrically distributed vectors is obtained. It extends the previously studied case of identically distributed vectors by Bourgain, Meyer, Milman and Pajor and solves a problem raised there. Another corollary asserts that any finite dimensional normed space has a “random cotype 2”.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02350-6
Efim Gluskin 1 ; Vitali Milman 1

1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel 
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Efim Gluskin; Vitali Milman. Randomizing properties of convex high-dimensional bodies and some geometric inequalities. Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 875-879. doi : 10.1016/S1631-073X(02)02350-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02350-6/

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[3] F. Barthe On a reverse form of the Brascamp–Lieb inequality, Invent. Math., Volume 134 (1998) no. 2, pp. 335-361

[4] H.J. Brascamp; E.H. Lieb Best constant in Young's inequality, its converse and its generalization to more than three functions, Adv. Math., Volume 20 (1976), pp. 151-173

[5] H.J. Brascamp; E.H. Lieb; J.M. Luttinger A general rearrangement inequality for multiple integrals, J. Func. Anal., Volume 17 (1974), pp. 227-237

[6] J. Bourgain; M. Meyer; V. Milman; A. Pajor On a geometric inequality (Y. Lindenstrauss; V.D. Milman, eds.), Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1317, Springer-Verlag, 1988

[7] M. Gromov; V.D. Milman A topological application of the isoperimetric inequality, Amer. J. Math., Volume 105 (1983), pp. 843-854

[8] V.D. Milman; G. Schechtman Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math., 1200, Springer-Verlag, 1986

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