An isomorphic Dvoretzky's theorem for convex bodies

Volume 127 / 1998

Y. Gordon, O. Guédon, M. Meyer Studia Mathematica 127 (1998), 191-200 DOI: 10.4064/sm-127-2-191-200


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.


  • Y. Gordon
  • O. Guédon
  • M. Meyer

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