Abstract
It is shown that the set [G,ϕ]Γ of immersed linear networks in \(\mathbb{R}^n \) that are parallel to a given immersed linear network \(\Gamma :G\to\mathbb{R}^n \) and have the same boundary ϕ as Γ is a convex polyhedral subset of the configuration space of movable vertices of the graph G. The dimension of [G,ϕ]Γ is calculated, and the number of its maximal faces is estimated. As an application, the spaces of all locally minimal and weighted minimal networks with fixed boundary and topology in \(\mathbb{R}^n \) are described. Bibliography: 21 titles.
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Ivanov, A.O., Tuzhilin, A.A. Linear Networks and Convex Polytopes. Journal of Mathematical Sciences 104, 1283–1288 (2001). https://doi.org/10.1023/A:1011377730044
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DOI: https://doi.org/10.1023/A:1011377730044