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 Mat. Sb., 2001, Volume 192, Number 6, Pages 31–50 (Mi msb571)

Differential calculus on the space of Steiner minimal trees in Riemannian manifolds

A. O. Ivanov, A. A. Tuzhilin

M. V. Lomonosov Moscow State University

Abstract: It is proved that the length of a minimal spanning tree, the length of a Steiner minimal tree, and the Steiner ratio regarded as functions of finite subsets of a connected complete Riemannian manifold have directional derivatives in all directions. The derivatives of these functions are calculated and some properties of their critical points are found. In particular, a geometric criterion for a finite set to be critical for the Steiner ratio is found. This criterion imposes essential restrictions on the geometry of the sets for which the Steiner ratio attains its minimum, that is, the sets on which the Steiner ratio of the boundary set is equal to the Steiner ratio of the ambient space.

DOI: https://doi.org/10.4213/sm571

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English version:
Sbornik: Mathematics, 2001, 192:6, 823–841

Bibliographic databases:

UDC: 514.77+512.816.4+517.924.8
MSC: Primary 05C05; Secondary 05C10, 05C35, 51M16, 57M15

Citation: A. O. Ivanov, A. A. Tuzhilin, “Differential calculus on the space of Steiner minimal trees in Riemannian manifolds”, Mat. Sb., 192:6 (2001), 31–50; Sb. Math., 192:6 (2001), 823–841

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb571
• https://doi.org/10.4213/sm571
• http://mi.mathnet.ru/eng/msb/v192/i6/p31

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. O. Ivanov, A. A. Tuzhilin, “Uniqueness of Steiner minimal trees on boundaries in general position”, Sb. Math., 197:9 (2006), 1309–1340
2. E. A. Zavalnyuk, “Steiner ratio for the Hadamard surfaces of curvature at most $k<0$”, J. Math. Sci., 203:6 (2014), 777–788
3. E. I. Stepanova, “Directional derivative of the weight of a minimal filling in Riemannian manifolds”, Moscow University Mathematics Bulletin, 70:1 (2015), 14–18
4. A. O. Ivanov, A. A. Tuzhilin, “Analiticheskie deformatsii minimalnykh setei”, Fundament. i prikl. matem., 21:5 (2016), 159–180
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