Differential calculus on the space of Steiner minimal trees in Riemannian manifolds
Journal Article
·
· Sbornik. Mathematics
- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
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.
- OSTI ID:
- 21205618
- Journal Information:
- Sbornik. Mathematics, Vol. 192, Issue 6; Other Information: DOI: 10.1070/SM2001v192n06ABEH000571; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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