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Mat. Sb., 1991, Volume 182, Number 12, Pages 1813–1844 (Mi msb1416)  

This article is cited in 18 scientific papers (total in 18 papers)

The Steiner problem in the plane or in plane minimal nets

A. O. Ivanov, A. A. Tuzhilin


Abstract: The famous Steiner problem in the Euclidean plane, which is that of investigating minimal nets spanning fixed finite subsets $M$ of points in the plane, is solved when $M$ is extremal, i.e. when $M$ lies on the boundary of its convex hull, and the nets are nondegenerate, i.e. have no vertices of degree 2.

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English version:
Mathematics of the USSR-Sbornik, 1993, 74:2, 555–582

Bibliographic databases:

MSC: 05B30, 90B06
Received: 21.11.1990

Citation: A. O. Ivanov, A. A. Tuzhilin, “The Steiner problem in the plane or in plane minimal nets”, Mat. Sb., 182:12 (1991), 1813–1844; Math. USSR-Sb., 74:2 (1993), 555–582

Citation in format AMSBIB
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\jour Math. USSR-Sb.
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\vol 74
\issue 2
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. O. Ivanov, A. A. Tuzhilin, “Geometry of minimal networks and the one-dimensional Plateau problem”, Russian Math. Surveys, 47:2 (1992), 59–131  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. O. Ivanov, I. V. Ptitsyna, A. A. Tuzhilin, “Classification of closed minimal networks on flat two-dimensional tori”, Russian Acad. Sci. Sb. Math., 77:2 (1994), 391–425  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. Ivanov A., Iskhakov I., Tuzhilin A., “Minimal Networks on Regular Polygons - Realization of Linear Tilings”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1993, no. 6, 77–80  mathscinet  zmath  isi
    4. A. O. Ivanov, A. A. Tuzhilin, “Topologies of locally minimal planar binary trees”, Russian Math. Surveys, 49:6 (1994), 201–202  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. A. O. Ivanov, “The geometry of plane locally minimal binary trees”, Sb. Math., 186:9 (1995), 1271–1301  mathnet  crossref  mathscinet  zmath  isi
    6. A. O. Ivanov, A. A. Tuzhilin, “On minimal binary trees with a regular boundary”, Russian Math. Surveys, 51:1 (1996), 144–145  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. A. O. Ivanov, A. A. Tuzhilin, “Structure of the set of planar minimal networks with given topology and boundary”, Russian Math. Surveys, 51:3 (1996), 553–554  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. A. O. Ivanov, A. A. Tuzhilin, “The classification of minimal skeletons with a regular boundary”, Russian Math. Surveys, 51:4 (1996), 734–735  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    9. A. A. Tuzhilin, “Minimal binary trees with regular boundary: the case of sceletons with four ends”, Sb. Math., 187:4 (1996), 581–622  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. A. O. Ivanov, A. A. Tuzhilin, “The twist number of planar linear trees”, Sb. Math., 187:8 (1996), 1149–1195  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. A. O. Ivanov, A. A. Tuzhilin, “The geometry of minimal networks with a given topology and a fixed boundary”, Izv. Math., 61:6 (1997), 1231–1263  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. A. A. Tuzhilin, “Minimal binary trees with a regular boundary: The case of skeletons with five endpoints”, Math. Notes, 61:6 (1997), 758–769  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. A. O. Ivanov, A. A. Tuzhilin, “Geometry of convex polygons and locally minimal binary trees spanning these polygons”, Sb. Math., 190:1 (1999), 71–110  mathnet  crossref  crossref  mathscinet  zmath  isi
    14. A. O. Ivanov, A. A. Tuzhilin, “The space of parallel linear networks with a fixed boundary”, Izv. Math., 63:5 (1999), 923–962  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. Ivanov, AO, “Extreme networks”, Acta Applicandae Mathematicae, 66:3 (2001), 251  crossref  mathscinet  zmath  isi  elib
    16. A. O. Ivanov, A. A. Tuzhilin, “Branching geodesics in normed spaces”, Izv. Math., 66:5 (2002), 905–948  mathnet  crossref  crossref  mathscinet  zmath
    17. N. S. Gusev, “Convex Realizations of Planar Linear Trees”, Math. Notes, 73:5 (2003), 625–635  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Ivanov A.O. Tuzhilin A.A., “Minimal Networks: a Review”, Advances in Dynamical Systems and Control, Studies in Systems Decision and Control, 69, ed. Sadovnichiy V. Zgurovsky M., Springer Int Publishing Ag, 2016, 43–80  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1991 Sbornik: Mathematics (from 1967)
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