Lipschitz distance between clouds
Ivan N. Mikhailov
TL;DR
The paper investigates the Lipschitz distance between clouds—moduli of metric spaces defined via finite Gromov--Hausdorff distance. It shows that the cloud of bounded metric spaces $[\Delta_1]$ and the cloud of the real line $[\mathbb{R}]$ have positive Lipschitz distance, by deriving a necessary and sufficient condition for $d_L(X,Y)=0$ in terms of bijections that are almost bi-Lipschitz. The result leverages the Lipschitz structure (continuous homotheties) to study cloud geometry and uses GH-distance machinery and correspondences to obtain quantitative bounds. The paper proves a concrete lower bound $d_L([\Delta_1],[\mathbb{R}]) \ge \ln(4/3)$ and discusses the open question of whether the distance is finite, contributing to the understanding of the global geometry of clouds under Lipschitz deformations.
Abstract
In this note we show that the Lipschitz distance between the classes of metric spaces at finite Gromov-Hausdorff distances from the one-point metric space and the real line with the natural metric, respectively, is positive.