Explicit geodesics in Gromov-Hausdorff space

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doi:10.3934/era.2018.25.006

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2018, Volume 25: 48-59. Doi: 10.3934/era.2018.25.006

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Explicit geodesics in Gromov-Hausdorff space

Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210. Phone: (614) 292-4975, Fax: (614) 292-1479

Received: March 30, 2017

Revised: March 12, 2018

Published: March 2018

This work was supported by NSF grants CCF-1526513 and IIS-1422400

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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 $n\geq 1$.

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Mathematics Subject Classification: Primary 53C23, Secondary 51F99.

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Figure 1. Branching geodesics as described in §1.1.2

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