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

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.

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