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Mat. Sb., 2012, Volume 203, Number 5, Pages 65–118 (Mi msb7777)  

This article is cited in 35 scientific papers (total in 35 papers)

One-dimensional Gromov minimal filling problem

A. O. Ivanovab, A. A. Tuzhilinba

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Laboratory of Discrete and Computational Geometry named after B. N. Delone of P. G. Demidov Yaroslavl State University

Abstract: The paper is devoted to a new branch in the theory of one-dimensional variational problems with branching extremals, the investigation of one-dimensional minimal fillings introduced by the authors. On the one hand, this problem is a one-dimensional version of a generalization of Gromov's minimal fillings problem to the case of stratified manifolds. On the other hand, this problem is interesting in itself and also can be considered as a generalization of another classical problem, the Steiner problem on the construction of a shortest network connecting a given set of terminals. Besides the statement of the problem, we discuss several properties of the minimal fillings and state several conjectures.
Bibliography: 38 titles.

Keywords: metric spaces, Gromov minimal fillings, Steiner minimal trees, minimal spanning trees, Steiner ratio.
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English version:
Sbornik: Mathematics, 2012, 203:5, 677–726

Bibliographic databases:

UDC: 514.774.8+515.124.4+519.176
MSC: Primary 05C12, 54E35; Secondary 05C05, 52A38
Received: 04.08.2010 and 08.04.2011

Citation: A. O. Ivanov, A. A. Tuzhilin, “One-dimensional Gromov minimal filling problem”, Mat. Sb., 203:5 (2012), 65–118; Sb. Math., 203:5 (2012), 677–726

Citation in format AMSBIB
\by A.~O.~Ivanov, A.~A.~Tuzhilin
\paper One-dimensional Gromov minimal filling problem
\jour Mat. Sb.
\yr 2012
\vol 203
\issue 5
\pages 65--118
\jour Sb. Math.
\yr 2012
\vol 203
\issue 5
\pages 677--726

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    This publication is cited in the following articles:
    1. A. O. Ivanov, Z. N. Ovsyannikov, N. P. Strelkova, A. A. Tuzhilin, “One-dimensional minimal fillings with negative edge weights”, Moscow University Mathematics Bulletin, 67:5-6 (2012), 189–194  mathnet  crossref  mathscinet
    2. Rubleva O.V., “Kriterii additivnosti konechnogo metricheskogo prostranstva i minimalnye zapolneniya”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2012, no. 2, 8–11  mathnet  mathscinet  zmath  elib
    3. N. Dolbilin, G. Edelsbrunner, A. Ivanov, O. Musin, “Pervaya yaroslavskaya letnyaya shkola po diskretnoi i vychislitelnoi geometrii”, Model. i analiz inform. sistem, 19:4 (2012), 168–173  mathnet
    4. H. Edelsbrunner, A. Ivanov, R. Karasev, “Current Open Problems in Discrete and Computational Geometry”, Model. i analiz inform. sistem, 19:5 (2012), 5–17  mathnet
    5. A. Yu. Eremin, “A formula for the weight of a minimal filling of a finite metric space”, Sb. Math., 204:9 (2013), 1285–1306  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. V. A. Mishchenko, “Estimates for the Steiner–Gromov ratio of Riemannian manifolds”, J. Math. Sci., 203:6 (2014), 833–836  mathnet  crossref  mathscinet  elib
    7. Z. N. Ovsyannikov, “An open family of sets that have several minimal fillings”, J. Math. Sci., 203:6 (2014), 855–857  mathnet  crossref  mathscinet  elib
    8. Z. N. Ovsyannikov, “The Steiner and Gromov–Steiner ratios and Steiner subratio in the space of compacta in the Euclidean plane with Hausdorff distance”, J. Math. Sci., 203:6 (2014), 858–863  mathnet  crossref  mathscinet  elib
    9. Z. N. Ovsyannikov, “The Steiner subratio of five points on a plane and four points in three-dimensional space”, J. Math. Sci., 203:6 (2014), 864–872  mathnet  crossref  mathscinet  elib
    10. V. N. Salnikov, “Probabilistic properties of topologies of finite metric spaces' minimal fillings”, J. Math. Sci., 203:6 (2014), 873–883  mathnet  crossref  mathscinet  elib
    11. Ivanov A.O. Tuzhilin A.A., “Gromov minimal fillings for finite metric spaces”, Publ. Inst. Math. (Beograd) (N.S.), 94:108 (2013), 3–15  crossref  mathscinet  zmath  isi  scopus
    12. B. B. Bednov, P. A. Borodin, “Banach spaces that realize minimal fillings”, Sb. Math., 205:4 (2014), 459–475  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. A. C. Pahkomova, “A Continuity Criterion for Steiner-Type Ratios in the Gromov–Hausdorff Space”, Math. Notes, 96:1 (2014), 130–139  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. A. C. Pahkomova, “Estimates of Steiner subratio and Steiner–Gromov ratio”, Moscow University Mathematics Bulletin, 69:1 (2014), 16–23  mathnet  crossref  elib
    15. “Minimal Fillings of Finite Metric Spaces: the State of the Art”, Discrete Geometry and Algebraic Combinatorics, Contemporary Mathematics, 625, 2014, 9+  crossref  isi
    16. E. I. Stepanova, “Directional derivative of the weight of a minimal filling in Riemannian manifolds”, Moscow University Mathematics Bulletin, 70:1 (2015), 14–18  mathnet  crossref  mathscinet  isi  elib
    17. S. Yu. Lipatov, “The functions that do not change types of minimal fillings”, Moscow University Mathematics Bulletin, 70:6 (2015), 267–269  mathnet  crossref  mathscinet  isi
    18. O. V. Rubleva, “Ricci curvature of a weighted tree”, Moscow University Mathematics Bulletin, 70:6 (2015), 278–279  mathnet  crossref  mathscinet  isi  elib
    19. A. O. Ivanov, A. A. Tuzhilin, “Branched coverings and Steiner ratio”, International Transactions in Operational Research, 23:5 (2016), 875–882  crossref  mathscinet  zmath  scopus
    20. B. B. Bednov, “The length of a minimal filling of star type”, Sb. Math., 207:8 (2016), 1064–1078  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. O. V. Rubleva, “The Ricci Curvature of a Weighted Tree”, Math. Notes, 100:4 (2016), 597–606  mathnet  crossref  crossref  mathscinet  isi  elib
    22. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
    23. A. O. Ivanov, A. A. Tuzhilin, “Minimal networks: a review”, Advances in dynamical systems and control, Stud. Syst. Decis. Control, 69, ed. V. Sadovnichiy, M. Zgurovsky, Springer, Cham, 2016, 43–80  crossref  mathscinet  zmath  isi  scopus
    24. E. I. Stepanova, “Bifurcations of Steiner minimal trees and minimal fillings for non-convex four-point boundaries and Steiner subratio for the Euclidean plane”, Moscow University Mathematics Bulletin, 71:2 (2016), 79–81  mathnet  crossref  mathscinet  isi
    25. A. O. Ivanov, A. A. Tuzhilin, “Analiticheskie deformatsii minimalnykh setei”, Fundament. i prikl. matem., 21:5 (2016), 159–180  mathnet
    26. A. S. Pakhomova, “Klassifikatsiya metricheskikh prostranstv, otnoshenie Shteinera—Gromova kotorykh ravno edinitse”, Fundament. i prikl. matem., 21:5 (2016), 181–189  mathnet
    27. S. Iliadis, A. O. Ivanov, A. A. Tuzhilin, “Local structure of Gromov–Hausdorff space, and isometric embeddings of finite metric spaces into this space”, Topology Appl., 221 (2017), 393–398  crossref  mathscinet  zmath  isi  scopus
    28. A. O. Ivanov, N. K. Nikolaeva, A. A. Tuzhilin, “Steiner's problem in the Gromov–Hausdorff space: the case of finite metric spaces”, Proc. Steklov Inst. Math. (Suppl.), 304, suppl. 1 (2019), S88–S96  mathnet  crossref  crossref  isi  elib
    29. B. B. Bednov, “The length of minimal filling for a five-point metric space”, Moscow University Mathematics Bulletin, 72:6 (2017), 221–225  mathnet  crossref  mathscinet  zmath  isi  elib
    30. L. Sh. Burusheva, “Banach spaces with shortest network length depending only on pairwise distances between points”, Sb. Math., 210:3 (2019), 297–309  mathnet  crossref  crossref  adsnasa  isi  elib
    31. Ivanov A. Tuzhilin A., “Steiner Type Ratios of Gromov-Hausdorff Space”, Eur. J. Comb., 80 (2019), 172–183  crossref  mathscinet  zmath  isi
    32. L. Burusheva, “The Steiner Subratio in Banach Spaces”, Math. Notes, 106:2 (2019), 183–190  mathnet  crossref  crossref  isi  elib
    33. Lipatov S.Yu., “Metrics Transformations Preserving the Types of One-Dimensional Minimal Fillings”, Filomat, 33:4, SI (2019), 1081–1089  crossref  mathscinet  isi
    34. E. I. Stepanova, “Bifurkatsii minimalnykh zapolnenii dlya chetyrëkh tochek evklidovoi ploskosti”, Fundament. i prikl. matem., 22:6 (2019), 253–261  mathnet
    35. A. Kh. Galstyan, “Problema Ferma—Shteinera v prostranstve kompaktnykh podmnozhestv evklidovoi ploskosti”, MaterialyXVII Vserossiiskoi molodezhnoishkoly-konferentsii Lobachevskie chteniya-2018,23-28 noyabrya 2018 g., Kazan.Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 175, VINITI RAN, M., 2020, 44–55  mathnet  crossref
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