On a distance characterization of A. D. Aleksandrov spaces of nonpositive curvature (Q960729)

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.