Coarse embedding (${\rho }_{-}(d_X)\le d_Y\le {\rho }_{+}(d_X)$)

A map $f:X\to Y$ between metric spaces is a coarse embedding when nondecreasing control functions ${\rho }_{-},{\rho }_{+}:[0,\infty )\to [0,\infty )$ satisfyThe last condition means that distances tending to infinity in the source must also tend to infinity in the target. The upper control prevents bounded source distances from having unbounded image distances. Unlike a metric embedding as defined here, a coarse embedding need not be injective: its lower control may vanish at small distances. Finite bi-Lipschitz distortion implies a coarse embedding by taking ${\rho }_{-}(t)=at$ and ${\rho }_{+}(t)=bt$, but nonlinear controls such as $\sqrt{t}$ allow much more compression.

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.