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1998 | 127 | 2 | 191-200
Tytuł artykułu

An isomorphic Dvoretzky's theorem for convex bodies

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We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in $ℝ^n$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of $ℝ^n$ satisfying $d(Y ∩ K, B_2^k) ≤ C(1+ √(k/ln(n/(kln(n+1))))$. This formulation of Dvoretzky's theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.
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autor
  • Equipe d'Analyse, Université de Marne-la-Vallée, 2 rue de la Butte Verte, 93166 Noisy-le-Grand Cedex, France , guedon@math.univ-mlv.fr
autor
  • Equipe d'Analyse, Université de Marne-la-Vallée, 2 rue de la Butte Verte, 93166 Noisy-le-Grand Cedex, France , meyer@math.univ-mlv.fr
Bibliografia
  • [BM] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $ℝ^n$, Invent. Math. 88 (1987), 319-340.
  • [DR] A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192-197.
  • [Go1] Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265-289.
  • [Go2] Y. Gordon, Majorization of gaussian processes and geometric applications, Probab. Theory Related Fields 91 (1992), 251-267.
  • [Go-M-P] Y. Gordon, M. Meyer and A. Pajor, Ratios of volumes and factorization through $ℓ ^∞$, Illinois J. Math. 40 (1996), 91-107.
  • [Gué] O. Guédon, Gaussian version of a theorem of Milman and Schechtman, Positivity 1 (1997), 1-5.
  • [M-S1] V. D. Milman and G. Schechtman, An "isomorphic" version of Dvoretzky's theorem, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 541-544.
  • [M-S2] V. D. Milman and G. Schechtman, An "isomorphic" version of Dvoretzky's theorem II, Math. Sci. Res. Inst. Publ., to appear.
  • [R1] M. Rudelson, Contact points of convex bodies, Israel J. Math., to appear.
  • [R2] M. Rudelson, Random vectors in the isotropic position, preprint.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv127i2p191bwm
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