We study some problems of geometrization of arbitrary metric spaces. In particular, we analyze the notions of straight and flat placements of points in these spaces. We continue the investigations of Kagan devoted to the detailed analysis of the notion of rectilinearity based on four groups of postulates. Our results 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 the rectilinear and flat placements of points in the metric space is investigated. Examples of placements of this kind based on linear functions in some classical spaces are presented. The presented results are obtained without using the property of completeness of the space and can be used for the discrete calculations and structuring of specific metric spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Savchenko and M. Zarichnyi, “Metrization of free groups on ultrametric spaces,” Topol. Appl., 157, No. 4, 724–729 (2010).
A. Savchenko and M. Zarichnyi, “Probability measure monad on the category of fuzzy ultrametric spaces,” Azerb. J. Math., 1, No. 1, 114–121 (2011).
A. D. Aleksandrov, Internal Geometry of Convex Surfaces [in Russian], Gostekhizdat, Moscow (1948).
V. F. Kagan, Foundations of Geometry [in Russian], Part 2, Gostekhizdat, Moscow (1956).
V. I. Kuz’mych, “Notion of angle in the investigation of the properties of metric spaces,” Visn. Cherkas. Univ. Ped. Nauky, No. 13, 26–32 (2016).
V. I. Kuz’mych, “Angle characteristic in metric spaces,” in: Abstr. of the Internat. Sci. Conf. “Algebraic and Geometric Methods of Analysis,” (May 31–June 5, 2017, Odessa) (2017), pp. 11–12.
V. I. Kuz’mych, “Construction of plane images in arbitrary metric spaces,” Visn. Cherkas. Univ. Ped. Nauky, No. 11, 40–46 (2017).
V. F. Kagan, Sketches of Geometry [in Russian], Moscow University, Moscow (1963).
V. I. Kuz’mych and Yu. V. Kuz’mych, “Analogs of the Jungius formula for the volume of tetrahedron,” Visn. Cherkas. Univ. Ped. Nauky, 249, No. 36, 55–64 (2012).
V. I. Kuz’mych, “Flatly placed sets of points in metric spaces,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 83, 58–71 (2017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 3, pp. 382–399, March, 2019.
Rights and permissions
About this article
Cite this article
Kuz’mych, V.I. Geometric Properties of Metric Spaces. Ukr Math J 71, 435–454 (2019). https://doi.org/10.1007/s11253-019-01656-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-019-01656-1