## Abstract

A new distance characterization of A.D. Aleksandrov domains of a space of nonpositive curvature has been provided. Aleksandrov domains were characterized by introducing a quasilinearization for an arbitrary metric space and formulating a Cauchy-Schwarz inequality, which implies an upper curvature bound of zero. A semimetric space with a quasi-inner product on it is called the quasilinearization of the semimetric space. It was proved that a semimetric space is isometric to a complete Aleksandrov domain if and only if it is weakly convex, each Cauchy sequence in this semimetric space has a limit, and it satisfies the four point cosq condition. It was also proved that a geodesically connected metric space is an Aleksandrov domain if and only if it satisfies the quadrilateral inequality condition.

Original language | English (US) |
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Pages (from-to) | 336-338 |

Number of pages | 3 |

Journal | Doklady Mathematics |

Volume | 75 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2007 |

## ASJC Scopus subject areas

- General Mathematics