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 CMFD: Year: Volume: Issue: Page: Find

 CMFD, 2013, Volume 51, Pages 5–20 (Mi cmfd251)

The length of an extremal network in a normed space: Maxwell formula

A. G. Bannikovaa, D. P. Ilyutkoab, I. M. Nikonovba

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.

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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 5--20 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd251} \transl \jour Journal of Mathematical Sciences \yr 2016 \vol 214 \issue 5 \pages 593--608 \crossref{https://doi.org/10.1007/s10958-016-2801-6}