Inequalities on Six Points in a $\mathrm{CAT}(0)$ Space

TL;DR

This paper introduces new inequalities applicable to any six points in a $ ext{CAT}(0)$ space, demonstrating they are independent of properties involving fewer points, thus revealing novel geometric constraints.

Contribution

It establishes a family of inequalities for six points in $ ext{CAT}(0)$ spaces that are independent of known inequalities for fewer points, expanding understanding of geometric relations.

Findings

Inequalities hold for any six points in $ ext{CAT}(0)$ spaces.

These inequalities are independent of 5-point and 4-point conditions.

The inequalities reveal new geometric constraints in $ ext{CAT}(0)$ spaces.

Abstract

We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{CAT}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{CAT}(0)$ spaces. In particular, the validity of these inequalities does not follow from the $\mathrm{CAT}(0)$ $4$-point condition.