Rough Isometries of Lipschitz Function Spaces

We show that rough isometries between metric spaces X, Y can be lifted to the spaces of real valued 1-Lipschitz functions over X and Y with supremum metric and apply this to their scaling limits. For the inverse, we show how rough isometries between X and Y can be reconstructed from structurally enriched rough isometries between their Lipschitz function spaces.

💡 Research Summary

The paper investigates the relationship between rough isometries (RIs) of metric spaces and the corresponding spaces of real‑valued 1‑Lipschitz functions equipped with the supremum metric. A rough isometry between two metric spaces (X) and (Y) is a map (\phi : X \to Y) that distorts distances by at most a fixed constant (\varepsilon) and whose image is (\varepsilon)-dense in (Y). The authors first show how any such (\varepsilon)-rough isometry can be “lifted’’ to a map between the Lipschitz function spaces (\operatorname{Lip}_1(X)) and (\operatorname{Lip}_1(Y)).

The lifting construction is explicit: for a given (f \in \operatorname{Lip}_1(X)) define
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