On a distance characterization of A.D. Aleksandrov spaces of nonpositive curvature

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.