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Classification of Metric Spaces Whose Steiner–Gromov Ratio is Equal to One


Several equivalent conditions for the Steiner–Gromov ratio of a metric space to be equal to one are stated, i.e., conditions for each minimal spanning tree in any finite subset of a given metric space to be both a shortest tree and a minimal filling. A complete classification of such spaces is obtained.

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  1. 1.

    M. Baronti, E. Casini, and P. L. Papini, “Equilateral sets and their central points,” Rend. Mat. Appl., 13, No. 1, 133–148 (1993).

    MathSciNet  MATH  Google Scholar 

  2. 2.

    B. B. Bednov and P. A. Borodin, “Banach spaces that realize minimal fillings,” Sb. Math., 205, No. 4, 459–475 (2014).

    MathSciNet  Article  Google Scholar 

  3. 3.

    P. A. Borodin, “An example of nonexistence of a Steiner point in a Banach space,” Math. Notes, 87, No. 3, 485–488 (2010).

    MathSciNet  Article  Google Scholar 

  4. 4.

    D. Cieslik, The Steiner Ratio, Kluwer Academic, Boston (2001).

    Book  Google Scholar 

  5. 5.

    A. L. Garkavi and V. A. Shmatkov, “On the Lamé point and its generalizations in a normed space,” Math. USSR-Sb., 24, No. 2, 267–286 (1974).

    Article  Google Scholar 

  6. 6.

    E. N. Gilbert and H. O. Pollak, “Steiner minimal trees,” SIAM J. Appl. Math., 16, No. 1, 1–29 (1968).

    MathSciNet  Article  Google Scholar 

  7. 7.

    M. Gromov, “Filling Riemannian manifolds,” J. Diff. Geom., 18, No. 1, 1–147 (1983).

    MathSciNet  Article  Google Scholar 

  8. 8.

    A. O. Ivanov and A. A. Tuzhilin, “One-dimensional Gromov minimal filling problem,” Sb. Math., 203, No. 5, 677–726 (2012).

    MathSciNet  Article  Google Scholar 

  9. 9.

    A. O. Ivanov and A. A. Tuzhilin, “Minimal fillings of finite metric spaces: The state of the art,” A. Barg and O. Musin, eds., Discrete Geometry and Algebraic Combinatorics, Contemp. Math., Vol. 625, Amer. Math. Soc., Providence (2014), pp. 9–35.

  10. 10.

    A. O. Ivanov and A. A. Tuzhilin, “The Steiner ratio Gilbert–Pollak conjecture is still open,” Algorithmica, 62, No. 1-2, 630–632 (2014).

  11. 11.

    A. S. Pakhomova, “A continuity criterion for Steiner-type ratios in the Gromov–Hausdorff space,” Math. Notes, 96, No. 1, 130–139 (2014).

    MathSciNet  Article  Google Scholar 

  12. 12.

    A. S. Pakhomova, “Estimates of Steiner subratio and Steiner–Gromov ratio,” Moscow Univ. Math. Bull., 69, No. 1, 16–23 (2014).

    MathSciNet  Article  Google Scholar 

  13. 13.

    P. L. Papini, “Two new examples of sets without medians and centers,” Soc. Estad. Invest. Operat. Top, 13, No. 2, 315–320 (2005).

    MathSciNet  MATH  Google Scholar 

  14. 14.

    B. Richmond and T. Richmond, “Metric spaces in which all triangles are degenerate,” Am. Math. Month., 104, No. 8, 713–719 (1997).

    MathSciNet  Article  Google Scholar 

  15. 15.

    L. Vesely, “A characterization of reflexivity in the terms of the existence of generalized centers,” Extr. Math., 8, No. 2-3, 125–131 (1993).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    K. A. Zareckii, “Constructing a tree on the basis of a set of distances between the hanging vertices,” Usp. Mat. Nauk, 20, No. 6, 90–92 (1965).

    MathSciNet  Google Scholar 

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Correspondence to A. S. Pakhomova.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 5, pp. 181–189, 2016.

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Pakhomova, A.S. Classification of Metric Spaces Whose Steiner–Gromov Ratio is Equal to One. J Math Sci 248, 636–641 (2020).

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