On the bifurcation of the solution of the Fermat–Steiner problem under $1$-parameter variation of the boundary in $H(\mathbb{R}^2)$

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 $\mathcal{A}$ is fixed in the space of non-empty closed and bounded subsets $H(X)$, then we will call the element $K \in H(X)$, at which the minimum of the sum of the distances to the elements of $\mathcal{A}$ is achieved, the Steiner astrovertex, the network connecting $\mathcal{A}$ with $K$ — the minimal astronet, and $\mathcal{A}$ 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 $H(\mathbb{R}^2)$, which illustrate geometric phenomena that are absent in the classical Steiner problem for points in $\mathbb{R}^2$, are studied.