Boundary stability in the Fermat--Steiner problem in hyperspaces over finite-dimensional normed spaces

TL;DR

This paper investigates the stability of solutions to the Fermat--Steiner problem in hyperspaces over finite-dimensional normed spaces, focusing on how convexification of boundary sets affects minimal distance sums.

Contribution

It introduces the concept of boundary stability in the Fermat--Steiner problem within hyperspaces and provides conditions for when boundary stability fails.

Findings

Convex hulls of boundary sets can alter minimal distance sums.

A sufficient condition for boundary instability was established.

The study advances understanding of geometric relationships in hyperspace Fermat--Steiner problems.

Abstract

The Fermat--Steiner problem is to find all points of the metric space Y such that the sum of the distances from each of them to points from some fixed finite subset A = {A_1, ..., A_n} of the space Y is minimal. This problem is considered in the case when Y=H(X) is the space of non-empty compact subsets of a finite-dimensional normed space X endowed with the Hausdorff metric, i.e. H(X) is a hyperspace over X. The set A is called boundary, all A_i are called boundary sets, and the compact sets that realize the minimum of the sum of distances to A_i are called Steiner compacts. In this paper, we study the question of stability in the Fermat--Steiner problem when passing from a boundary consisting of finite compact sets to a boundary consisting of their convex hulls. By stability here we mean that the minimum of the sum of distances does not change when passing to convex hulls of boundary compact sets. The paper continued the study of geometric objects, namely, hook sets that arise in the Fermat--Steiner problem. The results obtained revealed some geometry of the relationship between Steiner compacts and boundary sets. On its basis, a sufficient condition for the instability of the boundary in H(X) was derived.