On the bifurcation of the solution of the Fermat-Steiner problem under1-parameter variation of the boundary in (R2)
In this paper, we consider the Fermat-Steiner problem in hyperspaces with the Hausdorff metric. If is a metric space, and a non-empty finite subset is fixed in the space of nonempty closed and bounded subsets 𝐻(𝑋), then we will call the element ∈ 𝐻(𝑋), at which the minimum of the sum of the distances to the elements of is achieved, the Steiner astrovertex, the network connecting with - the minimal astronet, and 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.