Rough Isometries of Lipschitz Function Spaces

TL;DR

This paper explores how rough isometries between metric spaces can be extended to their Lipschitz function spaces and reconstructed from them, revealing deep structural connections.

Contribution

It introduces methods to lift and reconstruct rough isometries between metric spaces via their Lipschitz function spaces, advancing understanding of metric space isometries.

Findings

Rough isometries can be lifted to Lipschitz function spaces.

Rough isometries between spaces can be reconstructed from their function spaces.

Application to scaling limits of metric spaces.

Abstract

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.