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For two metric spaces X and Y, say that X {threshold-embeds} into Y if there exist a number K > 0 and a family of Lipschitz maps $f_{\tau} : X \to Y : \tau > 0 \}$ such that for every $x,y \in X$, \[ d_X(x,y) \geq \tau => d_Y(f_{\tau}(x),f_{\tau}(y)) \geq \|\varphi_{\tau}\|_{\Lip} \tau/K \] where $\|f_{\tau}\|_{\Lip}$ denotes the Lipschitz constant of $f_{\tau}$.
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