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, . . . , xn ∈ X there exists a linear mapping L : X → F, where F ⊆ X is a linear subspace of dimension O(log n), such that ‖xi − x j‖ ≤ ‖L(xi) − L(x j)‖ ≤ O(1) · ‖xi − x j‖ 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 22 O(log∗ n) . 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 En ⊆ Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.