 Uniform coarse embedding of a family of metric spaces

A family $(X_n)$ uniformly coarsely embeds into $Y$ when maps $f_n:X_n\to Y$ admit the same two control functions ${\rho }_{-}$ and ${\rho }_{+}$ from the definition of a coarse embedding, independent of $n$. For a single finite metric space, the lower control can vanish up to its diameter and then grow without bound; this says little about its representation. For a family with unbounded diameters, common controls have real content: they cannot be chosen afresh after seeing each member's diameter. The Hamming cube as an L1 and L2 metric supplies explicit common controls, in contrast with the expander graph obstruction to uniform coarse embedding.