Explicit Geodesics in Gromov-Hausdorff Space

TL;DR

This paper constructs explicit geodesics in the Gromov-Hausdorff space of compact metric spaces, providing a constructive proof of its geodesic nature and illustrating examples including spheres.

Contribution

It offers a new constructive proof that the Gromov-Hausdorff space is geodesic by explicitly constructing geodesics, with examples including spheres.

Findings

Gromov-Hausdorff space is a geodesic space.

Explicit geodesics can be constructed between various compact metric spaces.

Examples include geodesics between spheres of different dimensions.

Abstract

We provide an alternative, constructive proof that the collection $\mathcal{M}$ of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on $\mathcal{M}$. We also provide several interesting examples of geodesics on $\mathcal{M}$, including a geodesic between $\mathbb{S}^0$ and $\mathbb{S}^n$ for any $n\geq 1$.