Extended by Balk Metrics

  • DOVGOSHEY, OLEKSIY (Division of Applied Problems in Contemporary Analysis, Institute Mathematics of NASU, Donetsk National University) ;
  • DORDOVSKYI, DMYTRO (Institute of Applied Mathematics and Mechanics of NASU)
  • Received : 2013.12.21
  • Accepted : 2014.04.11
  • Published : 2015.06.23


Let X be a nonempty set and $\mathcal{F}$(X) be the set of nonempty finite subsets of X. The paper deals with the extended metrics ${\tau}:\mathcal{F}(X){\rightarrow}\mathbb{R}$ recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets A with ${\mid}A{\mid}{\leqslant}n$ make possible to consider "distance functions" with n variables and related them quantities. In particular, we study such type generalized diameters $diam_{{\tau}^n}$ and find conditions under which $B{\mapsto}diam_{{\tau}^n}B$ is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction ${\tau}$ to the set of $A{\in}\mathcal{F}(X)$ with ${\mid}A{\mid}{\leqslant}3$ is a symmetric G-metric. An infinitesimal analog for extended by Balk metrics is constructed.


  1. F. Abdullayev, O. Dovgoshey, M. Kucukaslan, Metric spaces with unique pretangent spaces. Conditions of the uniqueness, Ann. Acad. Sci. Fenn. Math., 36(2011), 353-392.
  2. P. I. Balk, Conseptual evaluation of ${\epsilon}$-equivalency in non-linear inverse gravity problems, Geophysical Journal, 31(6)(2009), 55-61.
  3. P. I. Balk, On an extension of the concept of metric (Russian), Dokl. Math., 79(3)(2009), 394-396.
  4. V. V. Bilet, Geodesic spaces tangent to metric spaces, Ukr. Math. J., 64(9)(2013), 1448-1456.
  5. V. Bilet, O. Dovgoshey, Isometric embeddings of pretangent spaces in $E^n$, Bull. Belg. Math. Soc. - Simon Stevin, 20(2013), 91-110.
  6. B. C. Dhage, Generalized metric space and mapping with fixed point, Bull. Calcutta Math. Soc., 84(4)(1992), 329-336.
  7. Dmitrii V. Dordovski, Metric tangent spaces to Euclidean spaces, J. Math. Sci. (N.Y.), 179(2)(2011), 229-244.
  8. D. Dordovskyi, O. Dovgoshey, E. Petrov, Diameter and Diametrical Pairs of Points in Ultrametric Spaces, p-Adic Numbers, Ultrametric Anal. App., 3(4)(2011), 253-262.
  9. O. Dovgoshey, Tangent spaces to metric spaces and to their subspaces, Ukr. Math. Bull., 5(4)(2008), 470-487.
  10. O. Dovgoshey, F. Abdullayev, M. Kucukaslan, Compactness and boundedness of tangent spaces to metric spaces, Beitrage Algebra Geom., 51(2)(2010), 547-576.
  11. A. A. Dovgoshey, D. V. Dordovskyi, An ultrametricity condition for pretangent spaces, Math. Notes, 92(7)(2012), 43-50.
  12. O. Dovgoshey, O. Martio, Tangent spaces to metric spaces, Reports in Math., Helsinki Univ., 480(2008), 1-20.
  13. O. Dovgoshey, O. Martio, Tangent spaces to general metric spaces, Rev. Roum. Math. Pures. Appl., 56(2)(2011), 137-155.
  14. S. Gahler, 2-metrische Raume und ihre topologische Structur, Mathematische Nachrichten, 26(1963), 115-148.
  15. S. Gahler, Zur geometric 2-metriche raume, Revue Roum. Math. Pures App., 40(1966), 664-669.
  16. J. L. Kelly, General Topology, D. Van Nostrand Company, 1965.
  17. P. Komjath, V. Totik, Problems and Theorems in Classical Set Theory, Springer, New York, 2006.
  18. Z. Mustafa, A new structure for generalized metric spaces - with applications to fixed point theory, Ph. D. thesis, The University of Newcastle, Callaghan, Australia, 2005.
  19. Z. Mustafa, O. Hamed, F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory App., 2008(2008), Article ID 189870, 12p. 1.
  20. Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7(2)(2006), 289-297.