Functions that Preserve Manhattan Distances

What functions, when applied to the pairwise Manhattan distances between any $n$ points, result in the Manhattan distances between another set of $n$ points? In this paper, we show that a function has this property if and only if it is Bernstein. This class of functions admits several classical analytic characterizations and includes $f(x) = x^s$ for $0 \leq s \leq 1$ as well as $f(x) = 1-e^{-xt}$ for any $t \geq 0$. While it was previously known that Bernstein functions had this property, it was not known that these were the only such functions. Our results are a natural extension of the work of Schoenberg from 1938, who addressed this question for Euclidean distances. Schoenberg's work has been applied in probability theory, harmonic analysis, machine learning, theoretical computer science, and more. We additionally show that if and only if $f$ is completely monotone, there exists \mbox{$F:\ell_1 \rightarrow \mathbb{R}^n$} for any $x_1, \ldots x_n \in \ell_1$ such that $f(\|x_i - x_j\|_1) = \langle F(x_i), F(x_j) \rangle$. Previously, it was known that completely monotone functions had this property, but it was not known they were the only such functions. The same result but with negative type distances instead of $\ell_1$ is the foundation of all kernel methods in machine learning, and was proven by Schoenberg in 1942.