Isometric embedding of all compact metric spaces into the Gromov-Hausdorff space

Determine whether every compact metric space admits an isometric embedding into the Gromov–Hausdorff space M, i.e., the metric space of isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.

Background

The paper studies embeddings into the Gromov–Hausdorff space M, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance. Prior work by Iliadis, Ivanov, and Tuzhilin established that all finite metric spaces can be isometrically embedded into M, leaving broader classes of spaces in question.

This paper proves that any bounded subset of a Euclidean space (equipped with the Chebyshev distance) can be isometrically embedded into M, which in turn implies bilipschitz embeddings for bounded, connected Riemannian manifolds. Despite these partial results, the general question for all compact metric spaces remains unresolved.