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Mat. Sb., 2017, Volume 208, Number 5, Pages 103–128 (Mi msb8751)  

Correlation between the norm and the geometry of minimal networks

I. L. Laut

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper is concerned with the inverse problem of the minimal Steiner network problem in a normed linear space. Namely, given a normed space in which all minimal networks are known for any finite point set, the problem is to describe all the norms on this space for which the minimal networks are the same as for the original norm. We survey the available results and prove that in the plane a rotund differentiable norm determines a distinctive set of minimal Steiner networks. In a two-dimensional space with rotund differentiable norm the coordinates of interior vertices of a nondegenerate minimal parametric network are shown to vary continuously under small deformations of the boundary set, and the turn direction of the network is determined.
Bibliography: 15 titles.

Keywords: Fermat point, minimal Steiner network, minimal parametric network, normed space, norm.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00378-а
Ministry of Education and Science of the Russian Federation НШ-7962.2016.1
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00378-a) and the Programme for the Support of Leading Scientific Schools of the President of the Russian Federation (grant no. НШ-7962.2016.1).


DOI: https://doi.org/10.4213/sm8751

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English version:
Sbornik: Mathematics, 2017, 208:5, 684–706

Bibliographic databases:

UDC: 514.77+519.176+517.982.22
MSC: Primary 05C35, 51M16; Secondary 05C05, 52B05
Received: 08.06.2016 and 18.10.2016

Citation: I. L. Laut, “Correlation between the norm and the geometry of minimal networks”, Mat. Sb., 208:5 (2017), 103–128; Sb. Math., 208:5 (2017), 684–706

Citation in format AMSBIB
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