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Sbornik: Mathematics, 2001, Volume 192, Issue 6, Pages 823–841
DOI: https://doi.org/10.1070/SM2001v192n06ABEH000571 (Mi sm571) 		 
This article is cited in 5 scientific papers (total in 5 papers)

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

A. O. Ivanov, A. A. Tuzhilin

M. V. Lomonosov Moscow State University
English version PDF (240 kB) Citations (5)
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DOI: https://doi.org/10.1070/SM2001v192n06ABEH000571
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.
Received: 21.08.2000
Russian version:
Matematicheskii Sbornik, 2001, Volume 192, Number 6, Pages 31–50
DOI: https://doi.org/10.4213/sm571
Bibliographic databases:
UDC: 514.77+512.816.4+517.924.8
MSC: Primary 05C05; Secondary 05C10, 05C35, 51M16, 57M15
Language: English
Original paper language: Russian
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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    • This publication is cited in the following 5 articles:
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    Russian version PDF:219
    English version PDF:18
    References:51
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