# An intrinsic characterization of five points in a $\mathrm{CAT}(0)$ space

## TL;DR

This paper characterizes five-point metric spaces that can be isometrically embedded into CAT(0) spaces using inequalities derived from four-point configurations, extending previous results for fewer points.

## Contribution

It introduces a new set of necessary and sufficient conditions for embedding small metric spaces into CAT(0) spaces, generalizing Gromov's inequalities and cycle conditions.

## Key findings

- Characterization of five-point spaces via $oxtimes$-inequalities.
- Equivalence of these inequalities to embeddability into CAT(0) spaces.
- Extension of Gromov's four-point inequalities to five points.

## Abstract

Gromov (2001) and Sturm (2003) proved that any four points in a $\mathrm{CAT}(0)$ space satisfy a certain family of inequalities. We call those inequalities the $\boxtimes$-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 $\mathrm{CAT}(0)$ space if and only if any four points in $X$ satisfy the $\boxtimes$-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a $\mathrm{CAT}(0)$ space.