## Abstract

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine-a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space (M,ρ) is an Aleksandrov R_{0} domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in M. We also observe that a geodesically connected metric space (M, ρ) is an R_{0} domain if and only if, for every quadruple of points in M, the quadrilateral inequality (known as Euler's inequality in ^{2}) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an R_{0}domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.

Original language | English (US) |
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Pages (from-to) | 195-218 |

Number of pages | 24 |

Journal | Geometriae Dedicata |

Volume | 133 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2008 |

## Keywords

- 2-Roundness
- Aleksandrov space
- Quadrilateral cosine
- Quadrilateral inequality
- ℛ domain

## ASJC Scopus subject areas

- Geometry and Topology