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Zap. Nauchn. Sem. POMI, 2001, Volume 279, Pages 111–140 (Mi znsl1456)  

This article is cited in 2 scientific papers (total in 2 papers)

Planar Manhattan local minimal and critical networks

A. O. Ivanova, V. L. Hongb, A. A. Tuzhilinc

a N. E. Bauman Moscow State Technical University
b Max Planck Institute for Mathematics in the Sciences
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: One-dimensional branching extremals of Lagrangian-type functionals are considered. Such extremals appear as a solutions to the classical Steiner problem on a shortest network, i.e., a connected system of paths that has smallest total length among all the networks spanning a given finite set of terminal points in the plane. In the present paper, the Manhattan-length functional is investigated, with Lagrangian equal to the sum of the absolute values of projections of the velocity vector onto the coordinate axes. Such functionals are useful in problems arising in Electronics, Robotics, chip desing, etc. In this case, in contrast to the case of the Steiner problem, local minimality does not imply extremality (however, each extreme network is locally minimal). A criterion of extremality is presented, which shows that the extrmality with respect to the Manhattan-length functional is a global topological property of networks.

Full text: PDF file (329 kB)

English version:
Journal of Mathematical Sciences (New York), 2004, 119:1, 55–70

Bibliographic databases:

UDC: 514.518
Received: 11.01.2001

Citation: A. O. Ivanov, V. L. Hong, A. A. Tuzhilin, “Planar Manhattan local minimal and critical networks”, Geometry and topology. Part 6, Zap. Nauchn. Sem. POMI, 279, POMI, St. Petersburg, 2001, 111–140; J. Math. Sci. (N. Y.), 119:1 (2004), 55–70

Citation in format AMSBIB
\Bibitem{IvaLeTuz01}
\by A.~O.~Ivanov, V.~L.~Hong, A.~A.~Tuzhilin
\paper Planar Manhattan local minimal and critical networks
\inbook Geometry and topology. Part~6
\serial Zap. Nauchn. Sem. POMI
\yr 2001
\vol 279
\pages 111--140
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1456}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1846075}
\zmath{https://zbmath.org/?q=an:1140.90338}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 119
\issue 1
\pages 55--70
\crossref{https://doi.org/10.1023/B:JOTH.0000008741.99645.42}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. G. Bannikova, D. P. Ilyutko, I. M. Nikonov, “The length of an extremal network in a normed space: Maxwell formula”, Journal of Mathematical Sciences, 214:5 (2016), 593–608  mathnet  crossref
    2. E. A. Zaval'nyuk, “Local structure of minimal networks in A. D. Alexandrov spaces”, Moscow University Mathematics Bulletin, 69:5 (2014), 220–224  mathnet  crossref  mathscinet
  • Записки научных семинаров ПОМИ
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