Geometric relations in an arbitrary metric space

V. I. Kuz’mich; A. G. Savchenko

doi:10.30970/ms.52.1.76-85

The paper is devoted to individual elements of metric geometry. Analytical relations are considered that concern the distances between points of a metric space and have a definite geometric interpretation. These relations are established on the basis of the concept of angle as an ordered triple of points of a metric space, its numerical characteristic, and are carried out in Euclidean geometry. It is shown that these relations are satised for points of an arbitrary metric space.

1. Kuz’mich V.I., The concept of angle in the study of the properties of a metric space, Visnyk Cherkaskoho Univ. Ser.: Ped. Nauky, 13 (2016), 26–32. (in Ukrainian)

2. Kuz’mich V.I., Angular characteristic in metric space, Algebr. and Geom. Methods of Analysis: Int. sci. Conf.: Abstracts, 2017, 11–12. (in Ukrainian)

3. Kuz’mich V.I., Flat placement set of points in a metric space, Visnyk Lviv Univ. Ser.: Mech.-Math., 83 (2017), 58–71. (in Ukrainian)

4. Kuz’mich V.I., Construction of flat images in an arbitrary metric space, Visnyk Cherkaskoho Univ. Ser.: Ped. Nauky, 11 (2017), 40–46. (in Ukrainian)

5. Kuz’mich V.I., Geometric properties of metric spaces, Ukr. Math. J., 71 (2019), №3, 382–399. (in Ukrainian)

6. Kagan V.F., The foundations of geometry. Part 2, M.-L.: Gostehizdat, 1956. (in Russian)

7. Kagan V.F., Essays on geometry, M.: Izdatel’stvo Moskovskogo universiteta, 1963. (in Russian)

8. Euclid, Elements. Books I-VI, M.-L.: Gostehizdat, 1948. (in Russian)

9. Hadamard J., Elementary geometry. Part 1, M.-L.: Gostehizdat, 1948. (in Russian)

10. Hilbert D., The foundations of geometry, Petrograd: Seyatel, 1923. (in Russian)

11. Aleksandrov A.D., Intrinsic geometry of convex surfaces, M.-L.: Hostehizdat, 1948. (in Russian)

12. Burago D., Burago Y., Ivanov S., A course in metric geometry, AMS: Providence, Rhode Island, 2001. 13. Berger M., Geometrie. Part 1, M.: Mir, 1984. (in Russian)

14. Ponarin Ya.P., Elementary geometry. Part 1, M.: Moscow Center for Continuous Math. Educ., 2004. (in Russian)

15. Ponarin Ya.P., Elementary geometry. Part 2, M.: Moscow Center for Continuous Math. Educ., 2006. (in Russian)

16. Kuz’mich V.I., Kuz’mich Yu.V. Analogs of formula Jungius volume tetrahedron, Visnyk Cherkaskoho Univ. Ser.: Ped. Nauky, 36(249) (2012), 55–64. (in Ukrainian)

	Geometric relations in an arbitrary metric space
Author	V. I. Kuz’mich¹, A. G. Savchenko² vikuzmichksu@gmail.com¹, savchenko.o.g@ukr.net² 1) Kherson State University, Kherson, Ukraine; 2) Kherson State Agrarian University, Kherson, Ukraine
Abstract	The paper is devoted to individual elements of metric geometry. Analytical relations are considered that concern the distances between points of a metric space and have a definite geometric interpretation. These relations are established on the basis of the concept of angle as an ordered triple of points of a metric space, its numerical characteristic, and are carried out in Euclidean geometry. It is shown that these relations are satised for points of an arbitrary metric space.
Keywords	metric space; angle in the metric space; straight-linearly placement of points of metric space; straight-linearly ordered placement of points of metric space; planar placement of points of metric space
DOI	doi:10.30970/ms.52.1.76-85
Reference	1. Kuz’mich V.I., The concept of angle in the study of the properties of a metric space, Visnyk Cherkaskoho Univ. Ser.: Ped. Nauky, 13 (2016), 26–32. (in Ukrainian) 2. Kuz’mich V.I., Angular characteristic in metric space, Algebr. and Geom. Methods of Analysis: Int. sci. Conf.: Abstracts, 2017, 11–12. (in Ukrainian) 3. Kuz’mich V.I., Flat placement set of points in a metric space, Visnyk Lviv Univ. Ser.: Mech.-Math., 83 (2017), 58–71. (in Ukrainian) 4. Kuz’mich V.I., Construction of flat images in an arbitrary metric space, Visnyk Cherkaskoho Univ. Ser.: Ped. Nauky, 11 (2017), 40–46. (in Ukrainian) 5. Kuz’mich V.I., Geometric properties of metric spaces, Ukr. Math. J., 71 (2019), №3, 382–399. (in Ukrainian) 6. Kagan V.F., The foundations of geometry. Part 2, M.-L.: Gostehizdat, 1956. (in Russian) 7. Kagan V.F., Essays on geometry, M.: Izdatel’stvo Moskovskogo universiteta, 1963. (in Russian) 8. Euclid, Elements. Books I-VI, M.-L.: Gostehizdat, 1948. (in Russian) 9. Hadamard J., Elementary geometry. Part 1, M.-L.: Gostehizdat, 1948. (in Russian) 10. Hilbert D., The foundations of geometry, Petrograd: Seyatel, 1923. (in Russian) 11. Aleksandrov A.D., Intrinsic geometry of convex surfaces, M.-L.: Hostehizdat, 1948. (in Russian) 12. Burago D., Burago Y., Ivanov S., A course in metric geometry, AMS: Providence, Rhode Island, 2001. 13. Berger M., Geometrie. Part 1, M.: Mir, 1984. (in Russian) 14. Ponarin Ya.P., Elementary geometry. Part 1, M.: Moscow Center for Continuous Math. Educ., 2004. (in Russian) 15. Ponarin Ya.P., Elementary geometry. Part 2, M.: Moscow Center for Continuous Math. Educ., 2006. (in Russian) 16. Kuz’mich V.I., Kuz’mich Yu.V. Analogs of formula Jungius volume tetrahedron, Visnyk Cherkaskoho Univ. Ser.: Ped. Nauky, 36(249) (2012), 55–64. (in Ukrainian)
Pages	86-95
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
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