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

• Authors: William B. Johnson;Assaf Naor
• Venue: SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
• Year: 2009

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.