A lower bound on dimension reduction for trees in \ell_1

There is a constant c > 0 such that for every $ϵ\in (0,1)$ and $n \geq 1/\epsilon^2$,thefollowingholds\mathrm{.}Anymappingfromthe$n$-point star metric into ${\mathrm{\ell }}_1^d$ with bi-Lipschitz distortion $1+ϵ$ requires dimension $$d\ge \frac{c\log n}{{ϵ}^2\log (1\mathrm{/}ϵ)}\mathrm{.}$$