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BY 4.0 license Open Access Published by De Gruyter Open Access August 27, 2020

An Intrinsic Characterization of Five Points in a CAT(0) Space

Tetsu Toyoda


Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.


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Received: 2019-08-18
Accepted: 2020-05-20
Published Online: 2020-08-27

© 2020 Tetsu Toyoda, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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