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 ∥x _i - x_j∥ ≤ ∥L(x_i) - L(x _j)∥ ≤ O(1)·∥x_i - 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 2^{2O(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 E_n ⊆ Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.