Part 1. General problems of the intellectual systems theory

Sufficient conditions for minimality of star networks in hyperspaces

A. M. Tropin

Lomonosov Moscow State University

Abstract: The Fermat-Steiner problem is to find a point in the metric space $ Y $ 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 minimum astronet. We consider this problem in the hyperspace $ Y = H(X) $ of nonempty, closed and bounded subsets of the proper metric space $ X $; moreover, the Hausdorff metric is introduced on $ H(X) $. Since $ X $ is proper space, all elements of $ H(X) $ are compact. Each solution of the Fermat-Steiner problem will be called the Steiner astrocompact; its set is divided into classes with equal weight, each of which corresponds to its own vector of distances to the boundary compact sets. In this article, we prove three sufficient conditions for the given compact set to be a Steiner astrocompact for a given boundary. Also, these conditions guarantee the uniqueness of the class of Steiner astrocompact spaces with equal weight. This theory is used to completely solve the Fermat-Steiner problem for some symmetric convex three-element boundaries in $ \mathbb {R}^2 $, and this is demonstrated by examples.

Keywords: Fermat-Steiner problem, star network, minimal astronet, Steiner astrocompact, hyperspace, Hausdorff distance, metric projection, point-to-set distance function, first variation.