Quasilinearization and curvature of Aleksandrov spaces

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 ${\left(\mathcal{M},\rho\right)}$ is an Aleksandrov ${\Re_{0}}$ 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 ${\mathcal{M}}$ . We also observe that a geodesically connected metric space ${\left(\mathcal{M},\rho\right)}$ is an ${\Re_{0}}$ domain if and only if, for every quadruple of points in ${\mathcal{M}}$ , the quadrilateral inequality (known as Euler’s inequality in ${\mathbb{R}^{2}}$) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an ${\Re_{0}}$ domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.