An Intrinsic Characterization of Five Points in a CAT(0) Space

Abstract

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

5-point Toponogov theorem

“…Since the associated quadratic form describes its metric completely, we may identify A 5 and B 5 with R 10 the space of quadratic forms on R 4 . This way we can think that A 5 and B 5 are convex cones in R 10 .…”

“…An analogous problem for 5-point sets in nonpositively curved spaces was solved by Tetsu Toyoda [10]; another solution is given in [5]. The 6-point case is open; Question 6.1 in [5] might be an answer.…”