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Mat. Sb., 2014, Volume 205, Number 4, Pages 3–20 (Mi msb8264)  

This article is cited in 10 scientific papers (total in 10 papers)

Banach spaces that realize minimal fillings

B. B. Bednov, P. A. Borodin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: It is proved that a real Banach space realizes minimal fillings for all its finite subsets (a shortest network spanning a fixed finite subset always exists and has the minimum possible length) if and only if it is a predual of $L_1$. The spaces $L_1$ are characterized in terms of Steiner points (medians).
Bibliography: 25 titles.

Keywords: Banach space, shortest network, minimal filling, Steiner point (median).

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00510
14-01-91158
Ministry of Education and Science of the Russian Federation НШ-3682.2014.1

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8264

Full text: PDF file (540 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2014, 205:4, 459–475

Bibliographic databases:

UDC: 517.982.256+515.124.4
MSC: Primary 46B04; Secondary 05C12, 54E35
Received: 19.06.2013 and 06.11.2013

Citation: B. B. Bednov, P. A. Borodin, “Banach spaces that realize minimal fillings”, Mat. Sb., 205:4 (2014), 3–20; Sb. Math., 205:4 (2014), 459–475

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. A. R. Alimov, “On finite-dimensional Banach spaces in which suns are connected”, Eurasian Math. J., 6:4 (2015), 7–18  mathnet
    3. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. 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
    5. A. R. Alimov, “Prostranstva Mazura i 4.3-svoistvo peresecheniya $(BM)$-prostranstv”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:2 (2016), 133–137  mathnet  crossref  mathscinet  elib
    6. A. S. Pakhomova, “Klassifikatsiya metricheskikh prostranstv, otnoshenie Shteinera—Gromova kotorykh ravno edinitse”, Fundament. i prikl. matem., 21:5 (2016), 181–189  mathnet
    7. A. O. Ivanov, N. K. Nikolaeva, A. A. Tuzhilin, “Problema Shteinera v prostranstve Gromova–Khausdorfa: sluchai konechnykh metricheskikh prostranstv”, Tr. IMM UrO RAN, 23, no. 4, 2017, 152–161  mathnet  crossref  elib
    8. 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
    9. 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  elib
    10. Ivanov A., Tuzhilin A., “Steiner Type Ratios of Gromov-Hausdorff Space”, Eur. J. Comb., 80 (2019), 172–183  crossref  isi
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