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Mat. Sb., 2013, Volume 204, Number 9, Pages 51–72 (Mi msb7835)  

This article is cited in 15 scientific papers (total in 15 papers)

A formula for the weight of a minimal filling of a finite metric space

A. Yu. Ereminab

a M. V. Lomonosov Moscow State University
b Delone Laboratory of Discrete and Computational Geometry, P. G. Demidov Yaroslavl State University

Abstract: We consider the problem of finding a minimal filling for a finite metric space, that is, a weighted graph of minimal weight joining a given finite metric space. We obtain a minimax formula for the weight of the minimal filling, which we use to prove various properties of minimal fillings.
Bibliography: 10 titles.

Keywords: minimal filling, finite metric spaces, graph, Gromov's problem, perimeter of a metric space.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00664а
Ministry of Education and Science of the Russian Federation НШ-1410.2012.1


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English version:
Sbornik: Mathematics, 2013, 204:9, 1285–1306

Bibliographic databases:

UDC: 515.124.4+519.176
MSC: Primary 05C35; Secondary 05C05, 05C22, 51K05
Received: 21.12.2010 and 07.05.2013

Citation: A. Yu. Eremin, “A formula for the weight of a minimal filling of a finite metric space”, Mat. Sb., 204:9 (2013), 51–72; Sb. Math., 204:9 (2013), 1285–1306

Citation in format AMSBIB
\by A.~Yu.~Eremin
\paper A formula for the weight of a~minimal filling of a~finite metric space
\jour Mat. Sb.
\yr 2013
\vol 204
\issue 9
\pages 51--72
\jour Sb. Math.
\yr 2013
\vol 204
\issue 9
\pages 1285--1306

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    This publication is cited in the following articles:
    1. 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
    2. 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
    3. 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
    4. 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
    5. 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
    6. A. O. Ivanov, A. A. Tuzhilin, “Minimal fillings of finite metric spaces: The state of the art”, Discrete geometry and algebraic combinatorics, Contemporary Mathematics, 625, eds. A. Barg, O. Musin, 2014, 9–35  crossref  mathscinet  zmath  isi
    7. A. C. Pahkomova, “Estimates of Steiner subratio and Steiner–Gromov ratio”, Moscow University Mathematics Bulletin, 69:1 (2014), 16–23  mathnet  crossref  elib
    8. 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
    9. 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
    10. 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
    11. A. O. Ivanov, A. A. Tuzhilin, “Minimal networks: a review”, Advances in dynamical systems and control, Stud. Syst. Decis. Control, 69, ed. V. A. Sadovnichiy, M. Z. Zgurovsky, Springer, [Cham], 2016, 43–80  crossref  mathscinet  zmath  isi  scopus
    12. A. O. Ivanov, A. A. Tuzhilin, “Analiticheskie deformatsii minimalnykh setei”, Fundament. i prikl. matem., 21:5 (2016), 159–180  mathnet
    13. 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
    14. 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
    15. E. I. Stepanova, “Bifurkatsii minimalnykh zapolnenii dlya chetyrëkh tochek evklidovoi ploskosti”, Fundament. i prikl. matem., 22:6 (2019), 253–261  mathnet
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