Geodesic in the Gromov-Hausdorff class for which the real line is an interior point

TL;DR

The paper addresses the existence of geodesics in the Gromov–Hausdorff class for unbounded metric spaces, proving a construction where the real line $\mathbb{R}$ sits as an interior point of a geodesic within the cloud $[\triangle_1]$. It introduces a canonical framework of clouds and leverages a gluing construction between a family of line-like spaces $\mathbb{Z}_t$ and an $\ell^1$-product space $\mathbb{R}_d$ to form a shortest curve in the GH-class. Key results include the explicit distance relations $d_{GH}(\mathbb{R}_{d_1},\mathbb{R}_{d_2})=\frac{|d_1-d_2|}{2}$ and $d_{GH}(\mathbb{Z}_{t_1},\mathbb{Z}_{t_2})=|t_1-t_2|$, together with a central bound $d_{GH}(\mathbb{Z}_t,\mathbb{R}_d)=\frac{d}{2}+\frac{1}{2}-t$ that underpins the geodesic construction. This extends the understanding of GH-geodesics beyond bounded clouds and demonstrates feasible interior-point behavior in the unbounded setting, contributing to the broader picture of the GH geometry of metric spaces.

Abstract

In this note we construct a geodesic line in the Gromov-Hausdorff class for which the real line with a natural metric is an interior point.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 1: BBI
• Theorem 1: BBI
• Definition 7
• Theorem 2: BBI