Geodesic
Correlation between the norm and the geometry of minimal networks
Sbornik. Mathematics, Tome 208 (2017) no. 5, pp. 684-706 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : norm. 
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     author = {I. L. Laut},
     title = {Correlation between the norm and the geometry of minimal networks},
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     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_5_a3/}
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I. L. Laut. Correlation between the norm and the geometry of minimal networks. Sbornik. Mathematics, Tome 208 (2017) no. 5, pp. 684-706. http://geodesic.mathdoc.fr/item/SM_2017_208_5_a3/

[1] A. O. Ivanov, A. A. Tuzhilin, “Branching geodesics in normed spaces”, Izv. Math., 66:5 (2002), 905–948 | DOI | DOI | MR | Zbl

[2] D. P. Il'yutko, “Branching extremals of the functional of $\lambda$-normed length”, Sb. Math., 197:5 (2006), 705–726 | DOI | DOI | MR | Zbl

[3] D. P. Il'yutko, “Locally minimal trees in $n$-normed spaces”, Math. Notes, 74:5 (2003), 619–629 | DOI | DOI | MR | Zbl

[4] D. P. Il'yutko, “Geometry of extremal nets on $\lambda$-normed planes”, Moscow Univ. Math. Bull., 60:4 (2005), 39–41 | MR | Zbl

[5] K. J. Swanepoel, “The local Steiner problem in finite-dimensional normed spaces”, Discrete Comput. Geom., 37:3 (2007), 419–442 | DOI | MR | Zbl

[6] C. Benítez, M. Fernández, M. L. Soriano, “Location of the Fermat–Torricelli medians of three points”, Trans. Amer. Math. Soc., 354:12 (2002), 5027–5038 | DOI | MR | Zbl

[7] I. L. Laut, Z. N. Ovsyannikov, “The type of minimal branching geodesics defines the norm in a normed space”, J. Math. Sci. (N. Y.), 203:6 (2014), 799–805 | DOI | MR | Zbl

[8] A. Kriegl, P. W. Michor, The convenient setting of global analysis, Math. Surveys Monogr., 53, Amer. Math. Soc., Providence, RI, 1997, x+618 pp. | DOI | MR | Zbl

[9] H. Martini, K. J. Swanepoel, G. Weiss, “The Fermat–Torricelli problem in normed planes and spaces”, J. Optim. Theory Appl., 115:2 (2002), 283–314 | DOI | MR | Zbl

[10] D. Cieslik, Steiner minimal trees, Nonconvex Optim. Appl., 23, Kluwer Acad. Publ., Dordrecht, 1998, xii+319 pp. | DOI | MR | Zbl

[11] H. Martini, K. J. Swanepoel, G. Weiss, “The geometry of Minkowski spaces – a survey. Part I”, Expo. Math., 19:2 (2001), 97–142 | DOI | MR | Zbl

[12] R. E. Megginson, An introduction to Banach space theory, Grad. Texts in Math., 183, Springer-Verlag, New York, 1998, xx+596 pp. | DOI | MR | Zbl

[13] A. O. Ivanov, A. A. Tuzhilin, “Stabilization of locally minimal trees”, Math. Notes, 86:4 (2009), 483–492 | DOI | DOI | MR | Zbl

[14] A. O. Ivanov, A. A. Tuzhilin, “The twist number of planar linear trees”, Sb. Math., 187:8 (1996), 1149–1195 | DOI | DOI | MR | Zbl

[15] E. N. Gilbert, H. O. Pollak, “Steiner minimal trees”, SIAM J. Appl. Math., 16:1 (1968), 1–29 | DOI | MR | Zbl