Geodesic
Uniqueness of Steiner minimal trees on boundaries
Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1309-1340 
A. O. Ivanov; A. A. Tuzhilin. Uniqueness of Steiner minimal trees on boundaries. Sbornik. Mathematics, Tome 197 (2006) no. 9, pp. 1309-1340. http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/
@article{SM_2006_197_9_a3,
     author = {A. O. Ivanov and A. A. Tuzhilin},
     title = {Uniqueness of {Steiner} minimal trees on boundaries},
     journal = {Sbornik. Mathematics},
     pages = {1309--1340},
     year = {2006},
     volume = {197},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/}
}
TY  - JOUR
AU  - A. O. Ivanov
AU  - A. A. Tuzhilin
TI  - Uniqueness of Steiner minimal trees on boundaries
JO  - Sbornik. Mathematics
PY  - 2006
SP  - 1309
EP  - 1340
VL  - 197
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/
LA  - en
ID  - SM_2006_197_9_a3
ER  - 
%0 Journal Article
%A A. O. Ivanov
%A A. A. Tuzhilin
%T Uniqueness of Steiner minimal trees on boundaries
%J Sbornik. Mathematics
%D 2006
%P 1309-1340
%V 197
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2006_197_9_a3/
%G en
%F SM_2006_197_9_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

The following result is proved: there exists an open dense subset $U$ of $\mathbb R^{2n}$ such that each $P\in U$ (regarded as an enumerated subset of the standard Euclidean plane $\mathbb R^2$) is spanned by a unique Steiner minimal tree, that is, a shortest non-degenerate network. Several interesting consequences are also obtained: in particular, it is proved that each planar Steiner tree is planar equivalent to a Steiner minimal tree. Bibliography: 11 titles.

[1] A. O. Ivanov, A. A. Tuzhilin, Minimal networks. The Steiner problem and its generalizations, CRC Press, Boca Raton, FL, 1994 | MR | Zbl

[2] A. O. Ivanov, A. A. Tuzhilin, Razvetvlennye geodezicheskie. Geometricheskaya teoriya lokalno minimalnykh setei, The Edwin Mellen Press, Lewiston, NY, 1999

[3] A. O. Ivanov, A. A. Tuzhilin, Branching solutions to one-dimensional variational problems, World Sci. Publ., River Edge, NJ, 2001 | MR | Zbl

[4] A. O. Ivanov, A. A. Tuzhilin, Teoriya ekstremalnykh setei, In-t kompyuternykh issledovanii, M., Izhevsk, 2003

[5] F. K. Hwang, D. Richards, P. Winter, The Steiner tree problem, Ann. Discrete Math., 53, North-Holland, Amsterdam, 1992 | MR | Zbl

[6] D. Cieslik, Steiner minimal trees, Kluwer Acad. Publ., Dordrecht, 1998 | MR | Zbl

[7] S. Hildebrandt, A. Tromba, The parsimonious universe, Copernicus, New York, 1996 | MR | Zbl

[8] Z. A. Melzak, Companion to concrete mathematics, Wiley, New York, 1973 | MR | Zbl

[9] A. O. Ivanov, A. A. Tuzhilin, “Differentsialnoe ischislenie na prostranstve minimalnykh derevev Shteinera v rimanovykh mnogoobraziyakh”, Matem. sb., 192:6 (2001), 31–50 | MR | Zbl

[10] A. O. Ivanov, A. A. Tuzhilin, D. Tsislik, “Otnoshenie Shteinera dlya rimanovykh mnogoobrazii”, UMN, 55:6 (2000), 139–140 | MR | Zbl

[11] A. O. Ivanov, A. A. Tuzhilin, D. Tsislik, “Otnoshenie Shteinera dlya mnogoobrazii”, Matem. zametki, 74:3 (2003), 387–395 | MR | Zbl