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Estimates for the Steiner–Gromov Ratio of Riemannian Manifolds

Abstract

The Steiner–Gromov ratio of a metric space X characterizes the ratio of the minimal filling weight to the minimal spanning tree length for a finite subset of X. It is proved that the Steiner–Gromov ratio of an arbitrary Riemannian manifold does not exceed the Steiner–Gromov ratio of the Euclidean space of the same dimension.

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References

  1. 1.

    N. Innami and B. H. Kim, “Steiner ratio for hyperbolic surfaces,” Proc. Jpn. Acad. Ser. A Math. Sci., 82, No. 6, 77–79 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    A. O. Ivanov and A. A. Tuzhilin, Extreme Networks Theory [in Russian], Izd. Inst. Komp. Issled., Izhevsk, Moscow (2003).

    Google Scholar 

  3. 3.

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

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    A. O. Ivanov, A. A. Tuzhilin, and D. Cieslik, “Steiner ratio for manifolds,” Math. Notes, 74, No. 3, 367–374 (2003).

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to V. A. Mishchenko.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 2, pp. 119–124, 2013.

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Mishchenko, V.A. Estimates for the Steiner–Gromov Ratio of Riemannian Manifolds. J Math Sci 203, 833–836 (2014). https://doi.org/10.1007/s10958-014-2173-8

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Keywords

  • Manifold
  • Riemannian Manifold
  • Span Tree
  • Steiner Tree
  • Weighted Graph