The Andoni-Naor-Neiman inequalities and isometric embeddability into a CAT(0) space

TL;DR

This paper demonstrates that a specific family of quadratic metric inequalities, valid in all CAT(0) spaces, does not characterize isometric embeddability into CAT(0) spaces by providing a counterexample.

Contribution

It shows that the family of inequalities by Andoni, Naor, and Neiman is not sufficient to characterize CAT(0) embeddability, using a counterexample metric space.

Findings

The 6-point Lebedeva metric space satisfies all inequalities.

This space does not embed into any CAT(0) space.

The inequalities do not fully characterize CAT(0) embeddability.

Abstract

Andoni, Naor and Neiman (2018) established a family of quadratic metric inequalities that hold true in every CAT(0) space. As stated in their paper, this family seems to include all previously used quadratic metric inequalities that hold true in every CAT(0) space. We prove that there exists a metric space that satisfies all inequalities in this family but does not admit an isometric embedding into any CAT(0) space. More precisely, we prove that the 6-point metric space constructed by Nina Lebedeva, which does not admit an isometric embedding into any CAT(0) space, satisfies all inequalities in this family.