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Zbl 1158.53034
Berg, I.D.; Nikolaev, I.G.
On a distance characterization of A. D. Aleksandrov spaces of nonpositive curvature. (English. Russian original)
[J] Dokl. Math. 75, No. 3, 336-338 (2007); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 414, No. 1, 10-12 (2007). ISSN 1064-5624; ISSN 1531-8362

In this paper, the authors give a new distance characterization of Aleksandrov $\Re_{0}$ domains of a space of curvature $\leq 0. $ The authors first prove that a geodesically connected metric space is an $ Re_{0}$ domain if and only if it satisfies the four point cosq condition. The four point cosq condition says that $\text{cosq}(AB, CD) \leq 1 $, where $$\text{cosq}(AB,CD)= \frac{\rho^2(A,D)+\rho^2(B,C)- \rho^29A,C)-\rho^2(B,D)} {2\rho(A,B)\rho(C,D)}$$ and $\rho$ is the metric for every pair of distinct points $(A,B)$ and $(C,D)$ in a semimetric space. Then they obtain the necessary and sufficient conditions for a semimetric space to be isometric to a complete $\Re_{0}$ domain. Finally, by using the averaging principle, they derive from the above results that a geodesically connected metric space is an $\Re_{0}$ domain if and only if it satisfies the quadrilateral inequality condition.
[Derong Qiu (Beijing)]

MSC 2000:: *53C23 Global topological methods (a la Gromov)

Keywords: cosq condition; semimetric space; geodesically connected metric space; $\Re_{0}$ domain

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