Branching geodesics of the Gromov--Hausdorff distance

TL;DR

This paper investigates the structure of the Gromov--Hausdorff space, analyzing topological distributions of special metric spaces and constructing branching geodesics that connect various classes of compact metric spaces.

Contribution

It introduces a continuous construction of branching geodesics in the Gromov--Hausdorff space, passing through diverse classes of metric spaces, revealing their geodesic and infinite-dimensional nature.

Findings

Sets of doubling, uniformly disconnected, and uniformly perfect spaces are topologically distributed within the Gromov--Hausdorff space.

Constructed branching geodesics pass through or avoid specific classes of spaces, illustrating the space's complex structure.

The Gromov--Hausdorff space contains topological embeddings of the Hilbert cube connecting any pair of compact metric spaces.

Abstract

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov--Hausdorff distance. We then construct branching geodesics of the Gromov--Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov--Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.