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Consider a function $f$ from reals to reals such that $f$, when applied to pairwise Manhattan distances between $n$ points, always results in a set of Manhattan distances.

Work by Schoenberg and Assouad show that any Bernstein function has this property (see Theorem 9.0.3 in the book “Geometry of cuts and metrics”, by Deza and Laurent). Recently, some collaborators and I showed that only Bernstein functions have this property. We published this on arXiv just now.

My question is: was this known before?

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    $\begingroup$ I was aware of the analogous result, when $\ell^1$-functions are replaced with conditionally negative definite functions. I'm not sure of a reference, possibly the book by Berg and Forst. $\endgroup$
    – YCor
    Nov 24, 2020 at 14:33

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