Abstract:
The properties of the Hausdorff mapping $\mathcal{H}$ taking each compact metric space to the space of its nonempty closed subspaces endowed with the Hausdorff metric are studied. It is shown that this mapping is nonexpanding (Lipschitz mapping with the constant $1$). Several examples of classes of metric spaces the distances between which are preserved by the mapping $\mathcal{H}$ are presented. The distance between any connected metric space with a finite diameter and any simplex with the greater diameter is calculated. Some properties of the Hausdorff mapping are discussed, which may help to understand whether the mapping $\mathcal{H}$ is isometric or not.
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