 The dependence on "ε" was studied by Yehoram Gordon, who showed that "k"*("X") ≥ "c"2 "ε"2 log "N".Another proof of this result was given by Gideon Schechtman.Noga Alon and Vitali Milman showed that the logarithmic bound on the dimension of the subspace in Dvoretzky's theorem can be significantly improved, if one is willing to accept a subspace that is close either to a Euclidean space or to a Chebyshev space.