= 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>.
Back to article page
