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Extreme networks. (English) Zbl 0990.05079
The paper gives an overview on the so-called extreme network theory. The origins of this theory are the classical Steiner problem, which asks to find a shortest network spanning a given finite set of points in the plane, and the one-dimensional variational calculus, which investigates extremals of variational functionals defined on a space of curves joining a pair of fixed points in an ambient space. As a common generalization of the two problems the paper discusses the structure of extreme networks in several spaces with a specific view to Riemannian manifolds, finite-dimensional real vector spaces and normalized spaces. A good collection of references is given.
MSC:
05C35 Extremal problems in graph theory
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
34B45 Boundary value problems on graphs and networks for ordinary differential equations
46B20 Geometry and structure of normed linear spaces
Citations:
Zbl 0983.05003
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