Geometric properties of metric spaces - Kuzmich, V. I. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
We study some problems of geometrization of arbitrary metric spaces. In particular, we studied the concept of straight andflat placement of points in this space. In a certain way, we continue the investigations of Kagan devoted to the detailedanalysis of the notion of straightforwardness based on four groups of postulates. The results of our work are based on thenotion of angular characteristics of three points of the space proposed by Alexandrov. We establish the conditions underwhich the set of points of an arbitrary metric space satisfies all five postulates of the first group of Kagan’s placementpostulates. The relationship between rectilinear and flat placements of points of the metric space is investigated. Examplesof placements of this kind based on linear functions in some classical spaces are presented. The results of the paperare obtained without using the property of completeness of the space and can be used for the discrete computation andstructuring of specific metric spaces.