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.

中文翻译：

---

CAT(0) 空间中五点的内在表征

摘要 Gromov (2001) 和 Sturm (2003) 证明了 CAT(0) 空间中的任何四个点都满足特定的不等式族。我们将这些不等式称为 ⊠ 不等式，遵循 Gromov 使用的符号。在本文中，我们证明了最多包含五个点的度量空间 X 允许等距嵌入到 CAT(0) 空间中，当且仅当 X 中的任何四个点满足 ⊠ 不等式。为了证明这一点，我们通过修改和概括 Gromov 循环条件，为度量空间引入了一系列新的必要条件，以允许将等距嵌入到 CAT(0) 空间中。此外，我们证明，如果度量空间满足所有这些必要条件，则它允许等距嵌入到 CAT(0) 空间中。