Lipschitz distance between clouds

TL;DR

This paper demonstrates that the Lipschitz distance between certain classes of metric spaces, specifically those close to a point or the real line in Gromov-Hausdorff terms, is always positive, highlighting a fundamental geometric property.

Contribution

It establishes a positive lower bound for the Lipschitz distance between classes of metric spaces near the point and real line in Gromov-Hausdorff space, revealing a new geometric insight.

Findings

Lipschitz distance is positive between classes near a point and the real line.

Shows a fundamental gap in the metric space of metric space classes.

Provides a new perspective on the structure of Gromov-Hausdorff space.

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.