
The length of an extremal network in a normed space: Maxwell formula
A. G. Bannikova^{a}, D. P. Ilyutko^{ab}, I. M. Nikonov^{ba} ^{a} Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Moscow, Russia
^{b} Delone Laboratory of Discrete and Computational Geometry, P. G. Demidov Yaroslavl State University, Yaroslavl, Russia
Abstract:
In the present paper we consider local minimal and extremal networks in normed spaces. It is well known that in the case of the Euclidean space these two classes coincide and the length of a local minimal network can be found by using only the coordinates of boundary vertices and the directions of boundary edges (the Maxwell formula). Moreover, as was shown by Ivanov and Tuzhilin [15], the length of a local minimal network in the Euclidean space can be found by using the coordinates of boundary vertices and the structure of the network. In the case of an arbitrary norm there are local minimal networks that are not extremal networks, and an analogue of the formula mentioned above is only true for extremal networks; this is the main result of the paper. Moreover, we generalize the Maxwell formula for the case of extremal networks in normed spaces and give an explicit construction of norming functionals used in the formula for several normed spaces.
Full text:
PDF file (678 kB)
References:
PDF file
HTML file
English version:
Journal of Mathematical Sciences, 2016, 214:5, 593–608
UDC:
514.77+519.711.72+517.982.22
Citation:
A. G. Bannikova, D. P. Ilyutko, I. M. Nikonov, “The length of an extremal network in a normed space: Maxwell formula”, Topology, CMFD, 51, PFUR, M., 2013, 5–20; Journal of Mathematical Sciences, 214:5 (2016), 593–608
Citation in format AMSBIB
\Bibitem{BanIlyNik13}
\by A.~G.~Bannikova, D.~P.~Ilyutko, I.~M.~Nikonov
\paper The length of an extremal network in a~normed space: Maxwell formula
\inbook Topology
\serial CMFD
\yr 2013
\vol 51
\pages 520
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd251}
\transl
\jour Journal of Mathematical Sciences
\yr 2016
\vol 214
\issue 5
\pages 593608
\crossref{https://doi.org/10.1007/s1095801628016}
Linking options:
http://mi.mathnet.ru/eng/cmfd251 http://mi.mathnet.ru/eng/cmfd/v51/p5
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  705  Full text:  65  References:  23 
