A lower bound on dimension reduction for trees in \ell_1

Abstract

There is a constant c > 0 such that for every $\epsilon \in (0,1)$ and $n \geq 1/\epsilon^2$, the following holds. Any mapping from the $n$-point star metric into $\ell_1^d$ with bi-Lipschitz distortion $1+\epsilon$ requires dimension $$d \geq {c\log n\over \epsilon^2\log (1/\epsilon)}.$$