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The Steiner Ratio Conjecture of Gilbert-Pollak May Still Be Open

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Abstract

We offer evidence in the disproof of the continuity of the length of minimum inner spanning trees with respect to a parameter vector having a zero component. The continuity property is the key step of the proof of the conjecture in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Therefore the Steiner ratio conjecture proposed by Gilbert-Pollak (SIAM J. Appl. Math. 16(1):1–29, 1968) has not been proved yet. The Steiner ratio of a round sphere has been discussed in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) by assuming the validity of the conjecture on a Euclidean plane in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Hence the results in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) have not been proved yet.

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References

  1. Du, D.Z., Hwang, F.K.: The Steiner ratio conjecture of Gilbert and Pollak is true. Proc. Nat. Acad. Sci. U.S.A. 87, 9464–9466 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Du, D.Z., Hwang, F.K.: A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica 7(1), 121–135 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gilbert, E.N., Pollak, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16(1), 1–29 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ivanov, A.O., Tuzhilin, A.A., Cieslik, D.: Steiner ratio for manifolds. Math. Notes 74(3), 367–374 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Innami, N., Kim, B.H.: Steiner ratio for hyperbolic surfaces. Proc. Jpn. Acad. Ser. A 82(6), 77–79 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Rubinstein, J.H., Weng, J.F.: Compression theorems and Steiner ratio on spheres. J. Comb. Optim. 1, 67–78 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan

    N. Innami

  2. Department of Mathematics, Kyung Hee University, Suwon, 446-701, Korea

    B. H. Kim

  3. Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

    Y. Mashiko

  4. Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, 8–19–1, Nanakuma, Jonan-ku, Fukuoka, 814-0180, Japan

    K. Shiohama

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  1. N. Innami
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  2. B. H. Kim
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  4. K. Shiohama
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Correspondence to N. Innami.

Additional information

The Research of B.H. Kim was supported by the Korea Research Council of Fundamental Science & Technology (KRCF) Grant No. C-RESEARCH-2006-11 NIMS.

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Innami, N., Kim, B.H., Mashiko, Y. et al. The Steiner Ratio Conjecture of Gilbert-Pollak May Still Be Open. Algorithmica 57, 869–872 (2010). https://doi.org/10.1007/s00453-008-9254-3

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  • DOI: https://doi.org/10.1007/s00453-008-9254-3

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