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On the bifurcation of the solution of the Fermat–Steiner problem under 1-parameter variation of the boundary in H(R2)
A. M. Tropin Lomonosov Moscow State University
(Moscow)
Abstract:
In this paper, we consider the Fermat–Steiner problem in hyperspaces with the Hausdorff metric. If X is a metric space, and a non-empty finite subset A is fixed in the space of non-empty closed and bounded subsets H(X), then we will call the element K∈H(X), at which the minimum of the sum of the distances to the elements of A is achieved, the Steiner astrovertex, the network connecting A with K — the minimal astronet, and A itself — the border. In the case of proper X, all its elements are compact, and the set of Steiner astrovertices is nonempty. In this article, we prove a criterion for when the Steiner astrovertex for one-point boundary compact sets in H(X) is one-point. In addition, a lower estimate for the length of the minimal parametric network is obtained in terms of the length of an astronet with one-point vertices contained in the boundary compact sets, and the properties of the boundaries for which an exact estimate is achieved are studied. Also bifurcations of Steiner astrovertices under 1-parameter deformation of three-element boundaries in H(R2), which illustrate geometric phenomena that are absent in the classical Steiner problem for points in R2, are studied.
Keywords:
Fermat–Steiner problem, Steiner minimal tree, minimal parametric network, minimal astronet, Steiner astrovertex, Steiner astrocompact, hyperspace, proper space, Hausdorff distance.
Received: 27.07.2021 Accepted: 06.12.2021
Citation:
A. M. Tropin, “On the bifurcation of the solution of the Fermat–Steiner problem under 1-parameter variation of the boundary in H(R2)”, Chebyshevskii Sb., 22:4 (2021), 265–288
Linking options:
https://www.mathnet.ru/eng/cheb1105 https://www.mathnet.ru/eng/cheb/v22/i4/p265
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Abstract page: | 101 | Full-text PDF : | 27 | References: | 26 |
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