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

A. Kh. Galstyan

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 81-128

Résumé

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 $A = \{A_1, \ldots, A_n\}$ of the space $Y$ is minimal. In this paper, this problem is considered in the case when $Y=\mathcal{H}(X)$ is the space of non-empty compact subsets of a finite-dimensional normed space $X$ endowed with the Hausdorff metric, i.e. $\mathcal{H}(X)$ 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 $A_i$ to a boundary consisting of their convex hulls $\mathrm{Conv}(A_i)$. By stability here we mean that the minimum of the sum of distances $S_A$ 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. Also three different sufficient conditions for the instability of the boundary from $\mathcal{H}(X)$ were derived, two of which are based on the constructed theory of such sets. For the case of an unstable boundary $A = \{A_1, \ldots, A_n\}$, a method was developed to search for deformations of some element from $\mathcal{H}(X),$ which lead to compact sets that give a smaller value of the sum of distances to $\mathrm{Conv}(A_i)$ than $S_A.$ The theory constructed within the framework of this study was applied to one of the well-known from recent works boundary $A\subset \mathcal{H}(\mathbb{R}^2),$ namely, its instability was proved and compact sets were found realizing the sum of distances to $\mathrm{Conv}(A_i),$ less than $S_A.$

Détail

Keywords: metric geometry, hyperspaces, convex sets, Hausdorff distance, Steiner problem, Fermat–Steiner problem, extremal networks.