Geometric properties of metric spaces

Abstract

We study some problems of geometrization of arbitrary metric spaces. In particular, we studied the concept of straight and flat placement of points in this space. In a certain way, we continue the investigations of Kagan devoted to the detailed analysis of the notion of straightforwardness based on four groups of postulates. The results of our work are based on the notion of angular characteristics of three points of the space proposed by Alexandrov. We establish the conditions under which the set of points of an arbitrary metric space satisfies all five postulates of the first group of Kagan’s placement postulates. The relationship between rectilinear and flat placements of points of the metric space is investigated. Examples of placements of this kind based on linear functions in some classical spaces are presented. The results of the paper are obtained without using the property of completeness of the space and can be used for the discrete computation and structuring of specific metric spaces.