Isometric Embeddings of Bounded Metric Spaces into the Gromov-Hausdorff Class

TL;DR

The paper proves that any bounded metric space can be embedded into the Gromov-Hausdorff class, using local geometric analysis and optimal correspondences, revealing new insights into the structure of GH.

Contribution

It introduces a method to isometrically embed any bounded metric space into the Gromov-Hausdorff class, based on local geometric descriptions and optimal correspondence techniques.

Findings

Any bounded metric space can be embedded into GH.

Local geometry of GH near generic spaces is characterized.

Optimal correspondences are effective in embedding proofs.

Abstract

It is shown that any bounded metric space can be isometrically embedded into the Gromov--Hausdorff metric class GH. This result is a consequence of local geometry description of the class GH in a sufficiently small neighborhood of a generic metric space. This description is interesting in itself. The technique of optimal correspondences and their distortions is used.