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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 ε X there exists a linear mapping L: X → F, where F ⊆ X is a linear subspace of dimension O(log n), such that||xi - xj|| ≤ ||L(xi) - L(xj)|| ≤ O(1) · ||xi - xj||for all i, j ε {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 [EQUATION]. 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.

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite