 Low-dimensional Frechet embedding into linfinity

For every integer $q\ge 2$, every $n$-point metric space embeds into ${\ell }_{\infty }^k$ with distortion at most $2q-1$ andThe coordinates are distance-to-subset functions. Randomly sampled subsets at $q$ density scales separate every pair at one scale; a union bound leaves the stated number of coordinates.