An isomorphic Dvoretzky's theorem for convex bodies

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 -