RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy Inst. Mat. i Mekh. UrO RAN:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 4, Pages 152–161 (Mi timm1475)  

Steiner's problem in the Gromov–Hausdorff space: the case of finite metric spaces

A. O. Ivanova, N. K. Nikolaevab, A. A. Tuzhilina

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, 119991 Russia
b SOSh NOU Orthodox Saint-Peter School, Moscow, 109028, Tessinskiy per., 3 Russia

Abstract: We study Steiner's problem in the Gromov–Hausdorff space, i.e., in the space of compact metric spaces (considered up to isometry) endowed with the Gromov-Hausdorff distance. Since this space is not boundedly compact, the problem of the existence of a shortest network connecting a finite point set in this space is open. We prove that each finite family of finite metric spaces can be connected by a shortest network. Moreover, it turns out that there exists a shortest tree all of whose vertices are finite metric spaces. A bound for the number of points in such metric spaces is derived. As an example, the case of three-point metric spaces is considered. We also prove that the Gromov-Hausdorff space does not realise minimal fillings, i.e., shortest trees in it need not be minimal fillings of their boundaries.

Keywords: Steiner's problem, shortest network, Steiner's minimal tree, minimal filling, Gromov-Hausdorff space, Gromov–Hausdorff distance.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00378
Ministry of Education and Science of the Russian Federation -7962.2016.1


DOI: https://doi.org/10.21538/0134-4889-2017-23-4-152-161

Full text: PDF file (221 kB)
References: PDF file   HTML file

Document Type: Article
UDC: 514+519.1
MSC: 58E10, 49K35, 05C35, 05C10, 30L05
Received: 23.06.2017

Citation: A. O. Ivanov, N. K. Nikolaeva, A. A. Tuzhilin, “Steiner's problem in the Gromov–Hausdorff space: the case of finite metric spaces”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 4, 2017, 152–161

Citation in format AMSBIB
\Bibitem{IvaNikTuz17}
\by A.~O.~Ivanov, N.~K.~Nikolaeva, A.~A.~Tuzhilin
\paper Steiner's problem in the Gromov--Hausdorff space: the case of finite metric spaces
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 4
\pages 152--161
\mathnet{http://mi.mathnet.ru/timm1475}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-4-152-161}
\elib{http://elibrary.ru/item.asp?id=30713969}


Linking options:
  • http://mi.mathnet.ru/eng/timm1475
  • http://mi.mathnet.ru/eng/timm/v23/i4/p152

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Trudy Instituta Matematiki i Mekhaniki UrO RAN
    Number of views:
    This page:45
    Full text:4
    References:6
    First page:6

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2018