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Geometric Properties of Metric Spaces

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Ukrainian Mathematical Journal Aims and scope

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.

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Authors and Affiliations

  1. Kherson State University, Kherson, Ukraine

    V. I. Kuz’mych

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  1. V. I. Kuz’mych
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Correspondence to V. I. Kuz’mych.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 3, pp. 382–399, March, 2019.

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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

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  • DOI: https://doi.org/10.1007/s11253-019-01656-1

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