The Andoni-Naor-Neiman Inequalities and Isometric Embeddability into a CAT(0) Space

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