An estimate for the length of a minimal parametric network in hyperspaces under deformation of the boundary set

The Fermat-Steiner problem is to find a point in the metric space Y (which we will call the Steiner astrovertex) such that the sum of the distances from it to the points of some finite fixed subset $ A \subset Y $, called the boundary, is minimal. We will call the minimal sum of distances the length of the minimal astronet. We consider this problem in the hyperspace $ Y = H(X) $ of nonempty, closed, and bounded subsets of the proper space X, which are compact in this space. This article describes a wide class of deformations of boundary compact sets that do not increase the length of the minimal astronet. Averaging in the sense of the Minkowski sum of a finite number of boundaries consisting of an equal number of elements is also considered, and it is shown that such averaging also does not increase the length of the minimal astronet.