The Hausdorff Mapping Is Nonexpanding

TL;DR

This paper studies the Hausdorff mapping, showing it is nonexpanding and exploring conditions under which it preserves distances, with implications for understanding its isometric properties.

Contribution

It proves that the Hausdorff mapping is nonexpanding and provides examples and calculations related to its distance-preserving properties.

Findings

Hausdorff mapping is nonexpanding (Lipschitz with constant 1)

Identifies classes of metric spaces with preserved distances under the mapping

Calculates distances between connected spaces and simplices with larger diameters

Abstract

In the present paper we investigate the properties of the Hausdorff mapping $\mathcal{H}$, which takes each compact metric space to the space of its nonempty closed subspaces. It is shown that this mapping is nonexpanding (Lipschitz mapping with constant $1$). This paper gives several examples of classes of metric spaces, distances between which are preserved by the mapping $\mathcal{H}$. We also calculate distance between any connected metric space and any simplex with greater diameter than the former one. At the end of the paper we discuss some properties of the Hausdorff mapping which may help to prove that it is isometric