Geodesic
On the bifurcation of the solution of the Fermat–Steiner problem under $1$-parameter variation of the boundary in $H(\mathbb{R}^2)$
Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 265-288 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we consider the Fermat–Steiner problem in hyperspaces with the Hausdorff metric. If $X$ is a metric space, and a non-empty finite subset $\mathcal{A}$ is fixed in the space of non-empty closed and bounded subsets $H(X)$, then we will call the element $K \in H(X)$, at which the minimum of the sum of the distances to the elements of $\mathcal{A}$ is achieved, the Steiner astrovertex, the network connecting $\mathcal{A}$ with $K$ — the minimal astronet, and $\mathcal{A}$ itself — the border. In the case of proper $X$, all its elements are compact, and the set of Steiner astrovertices is nonempty. In this article, we prove a criterion for when the Steiner astrovertex for one-point boundary compact sets in $H(X)$ is one-point. In addition, a lower estimate for the length of the minimal parametric network is obtained in terms of the length of an astronet with one-point vertices contained in the boundary compact sets, and the properties of the boundaries for which an exact estimate is achieved are studied. Also bifurcations of Steiner astrovertices under $1$-parameter deformation of three-element boundaries in $H(\mathbb{R}^2)$, which illustrate geometric phenomena that are absent in the classical Steiner problem for points in $\mathbb{R}^2$, are studied.
Keywords: Fermat–Steiner problem, Steiner minimal tree, minimal parametric network, minimal astronet, Steiner astrovertex, hyperspace, proper space
Mots-clés : Steiner astrocompact, Hausdorff distance. 
@article{CHEB_2021_22_4_a13,
     author = {A. M. Tropin},
     title = {On the bifurcation of the solution of the {Fermat{\textendash}Steiner} problem under $1$-parameter variation of the boundary in $H(\mathbb{R}^2)$},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {265--288},
     year = {2021},
     volume = {22},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_4_a13/}
}
TY  - JOUR
AU  - A. M. Tropin
TI  - On the bifurcation of the solution of the Fermat–Steiner problem under $1$-parameter variation of the boundary in $H(\mathbb{R}^2)$
JO  - Čebyševskij sbornik
PY  - 2021
SP  - 265
EP  - 288
VL  - 22
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CHEB_2021_22_4_a13/
LA  - ru
ID  - CHEB_2021_22_4_a13
ER  - 
%0 Journal Article
%A A. M. Tropin
%T On the bifurcation of the solution of the Fermat–Steiner problem under $1$-parameter variation of the boundary in $H(\mathbb{R}^2)$
%J Čebyševskij sbornik
%D 2021
%P 265-288
%V 22
%N 4
%U http://geodesic.mathdoc.fr/item/CHEB_2021_22_4_a13/
%G ru
%F CHEB_2021_22_4_a13
A. M. Tropin. On the bifurcation of the solution of the Fermat–Steiner problem under $1$-parameter variation of the boundary in $H(\mathbb{R}^2)$. Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 265-288. http://geodesic.mathdoc.fr/item/CHEB_2021_22_4_a13/

[1] Jarnik V., Kössler M., “O minimalnich grafech, obsahujicich n danych bodu”, Casopis pro pestovani matematiky a fysiky, 63:8 (1934), 223–235 | DOI | Zbl

[2] Cieslik D., $k$-Steiner minimal trees in metric spaces, Ernst–Moritz–Arndt–Univ. Greifswald, Inst. fur Mathematik und Informatik, 1999 | MR

[3] Ivanov A. O., Tuzhilin A. A., Minimal Networks. The Steiner Problems and Its Generalizations, CRC Press, Boca Raton, Fl, 1994 | MR

[4] Tuzhilin A. A., Fomenko A. T., “Mnogoznachnye otobrazheniya, minimalnye poverkhnosti i mylnye plenki”, Vestnik MGU, ser. matem., 1986, no. 3, 3–12 | MR

[5] Stepanova E. I., “Bifurkatsii minimalnykh derevev Shteinera i minimalnykh zapolnenii dlya nevypuklykh chetyrekhtochechnykh granits i subotnoshenie Shteinera evklidovoi ploskosti”, Vestn. Mosk. Un-ta., Ser. 1, Matematika. Mekhanika, 71:2 (2016), 48–51 | Zbl

[6] Stepanova E. I., “Bifurkatsii topologii derevev Shteinera na ploskosti”, Fund. i prikl. matem., 21:6 (2016), 183–204

[7] Arnold V. I., “Teoriya katastrof”, Dinamicheskie sistemy — 5, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 5, VINITI, M., 1986, 219–277

[8] Hausdorff F., Grundzüge der Mengenlehre, Veit, Leipzig, 1914 | MR

[9] Schlicker S., The geometry of the Hausdorff metric, GVSU REU, Grand Valley State Univ., Allendale, MI, 2008

[10] Ivanov A. O., Tuzhilin A. A., Geometriya rasstoyanii Khausdorfa i Gromova—Khausdorfa: sluchai kompaktov, Izd-vo Popechitelskogo soveta mekh-mat f-ta MGU, M., 2017

[11] Blackburn C. C., Lund K., Schlicker S., Sigmon P., Zupan A., An introduction to the geometry of $H(\mathbb{R}^n)$, GVSU REU, Grand Valley State Univ., Allendale, MI, 2007

[12] Emelichev V. A., Melnikov O. I., Sarvanov V. I., Tyshkevich R. I., Lektsii po teorii grafov, Nauka, M., 1990 | MR

[13] Burago D. Yu., Burago Yu. D., Ivanov S. V., Kurs metricheskoi geometrii, Izd-vo Instituta kompyuternykh issledovanii, M.—Izhevsk, 2004 | MR

[14] Ivanov A., Tropin A., Tuzhilin A., “Fermat—Steiner problem in the metric space of compact sets endowed with Hausdorff distance”, J. Geom., 108 (2017), 575–590 | DOI | MR | Zbl

[15] Galstyan A. Kh., Ivanov A. O., Tuzhilin A. A., “Problema Ferma — Shteinera v prostranstve kompaktnykh podmnozhestv $\mathbb{R}^m$ s metrikoi Khausdorfa”, Matem. sb., 212:1 (2021), 28–62 | DOI | MR | Zbl

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