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.
Tetsu Toyoda (Fri,) studied this question.