# An isomorphic Dvoretzky's theorem for convex bodies

Y. Gordon; O. Guédon; M. Meyer

Studia Mathematica (1998)

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

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topGordon, 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

top- [BM] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in ${\mathbb{R}}^{n}$, Invent. Math. 88 (1987), 319-340. Zbl0617.52006
- [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
- [Go1] Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265-289. Zbl0663.60034
- [Go2] Y. Gordon, Majorization of gaussian processes and geometric applications, Probab. Theory Related Fields 91 (1992), 251-267. Zbl0744.60039
- [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
- [Gué] O. Guédon, Gaussian version of a theorem of Milman and Schechtman, Positivity 1 (1997), 1-5. Zbl0912.46008
- [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
- [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
- [R1] M. Rudelson, Contact points of convex bodies, Israel J. Math., to appear. Zbl0896.52008
- [R2] M. Rudelson, Random vectors in the isotropic position, preprint. Zbl0929.46021

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