The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite

Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x₁,...,x_n∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that {double pipe}x_i-x_j{double pipe}≤{double pipe}L(x_i)-L(x_j){double pipe}≤O(1){dot operator}{double pipe}x_i-x_j{double pipe} for all i,j∈{1,...,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion,. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n, there exists an n-dimensional subspace E_n⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.