# An isomorphic Dvoretzky's theorem for convex bodies

Studia Mathematica (1998)

• Volume: 127, Issue: 2, page 191-200
• ISSN: 0039-3223

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## Abstract

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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 $d\left(Y\cap K,{B}_{2}^{k}\right)\le C\left(1+\surd \left(k/ln\left(n/\left(kln\left(n+1\right)\right)\right)\right)$. 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.

## How to cite

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Gordon, Y., Guédon, O., and Meyer, M.. "An isomorphic Dvoretzky's theorem for convex bodies." Studia Mathematica 127.2 (1998): 191-200. <http://eudml.org/doc/216466>.

@article{Gordon1998,
abstract = {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.},
author = {Gordon, Y., Guédon, O., Meyer, M.},
journal = {Studia Mathematica},
keywords = {Dvoretzky's theorem; convex bodies; Banach-Mazur distance; probabilistic Gaussian estimates},
language = {eng},
number = {2},
pages = {191-200},
title = {An isomorphic Dvoretzky's theorem for convex bodies},
url = {http://eudml.org/doc/216466},
volume = {127},
year = {1998},
}

TY - JOUR
AU - Gordon, Y.
AU - Guédon, O.
AU - Meyer, M.
TI - An isomorphic Dvoretzky's theorem for convex bodies
JO - Studia Mathematica
PY - 1998
VL - 127
IS - 2
SP - 191
EP - 200
AB - 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.
LA - eng
KW - Dvoretzky's theorem; convex bodies; Banach-Mazur distance; probabilistic Gaussian estimates
UR - http://eudml.org/doc/216466
ER -

## References

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1. [BM] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in ${ℝ}^{n}$, Invent. Math. 88 (1987), 319-340. Zbl0617.52006
2. [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. Zbl0036.36303
3. [Go1] Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265-289. Zbl0663.60034
4. [Go2] Y. Gordon, Majorization of gaussian processes and geometric applications, Probab. Theory Related Fields 91 (1992), 251-267. Zbl0744.60039
5. [Go-M-P] Y. Gordon, M. Meyer and A. Pajor, Ratios of volumes and factorization through ${\ell }^{\infty }$, Illinois J. Math. 40 (1996), 91-107. Zbl0843.46008
6. [Gué] O. Guédon, Gaussian version of a theorem of Milman and Schechtman, Positivity 1 (1997), 1-5. Zbl0912.46008
7. [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. Zbl0836.46007
8. [M-S2] V. D. Milman and G. Schechtman, An "isomorphic" version of Dvoretzky's theorem II, Math. Sci. Res. Inst. Publ., to appear. Zbl0942.46012
9. [R1] M. Rudelson, Contact points of convex bodies, Israel J. Math., to appear. Zbl0896.52008
10. [R2] M. Rudelson, Random vectors in the isotropic position, preprint. Zbl0929.46021

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