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

@article{Johnson2009TheJL,
  title={The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite},
  author={William B. Johnson and Assaf Naor},
  journal={Discrete & Computational Geometry},
  year={2009},
  volume={43},
  pages={542-553}
}
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… CONTINUE READING
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