On the bifurcation of the solution of the Fermat–Steiner problem under 1-parameter variation of the boundary in 𝐻(R2)
https://doi.org/10.22405/2226-8383-2021-22-4-265-288
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 nonempty
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.
About the Author
Alexander Mikhailovich Tropin
Lomonosov Moscow State University
Russian Federation
Russian Federation
graduate student
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Review
For citations:
Tropin A.M. On the bifurcation of the solution of the Fermat–Steiner problem under 1-parameter variation of the boundary in 𝐻(R2). Chebyshevskii Sbornik. 2021;22(4):265-288. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-4-265-288
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