The Fermat-Steiner problem in the space of compact subsets of endowed with the Hausdorff metric,Sbornik: Mathematics

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The Fermat-Steiner problem in the space of compact subsets of endowed with the Hausdorff metric
Sbornik: Mathematics ( IF 0.8 ) Pub Date : 2021-03-09 , DOI: 10.1070/sm9343
A. Kh. Galstyan _{1,

2} , A. O. Ivanov _{1,

2,

3} , A. A. Tuzhilin _{1,

2}

Affiliation

The Fermat-Steiner problem consists in finding all points in a metric space $X$ at which the sum of the distances to fixed points $A_1,\dots,A_n$ of $X$ attains its minimum value. This problem is studied in the metric space $\mathscr{H}(\mathbb R^m)$ of all nonempty compact subsets of the Euclidean space $\mathbb R^m$ , and the $A_i$ are pairwise disjoint finite sets in $\mathbb R^m$ . The set of solutions of this problem (which are called Steiner compact sets) falls into different classes in accordance with the distances to the $A_i$ . Each class contains an inclusion-greatest element and inclusion-minimal elements (a maximal Steiner compact set and minimal Steiner compact sets, respectively). We find a necessary and sufficient condition for a compact set to be a minimal Steiner compact set in a given class, provide an algorithm for constructing such compact sets and find a sharp estimate for their cardinalities. We also put forward a number of geometric properties of minimal and maximal compact sets. The results obtained can significantly facilitate the solution of specific problems, which is demonstrated by the well-known example of a symmetric set $\{A_1,A_2,A_3\}\subset \mathbb R^2$ , for which all Steiner compact sets are asymmetric. The analysis of this case is significantly simplified due to the technique developed.

Bibliography 16 titles.

中文翻译：

赋有 Hausdorff 度量的紧子集空间中的 Fermat-Steiner 问题

Fermat-Steiner 问题在于在一个度量空间中找到所有点， $X$ 在这些点到的固定点的距离之 $A_1，\点，A_N$ 和 $X$ 达到其最小值。这个问题是在 $\ mathscr {H}（\ mathbb R 1米）$ 欧几里德空间的所有非空紧子集的度量空间中研究的 $\ mathbb R 1米$ ，并且是 $A_i$ 中的成对不相交的有限集 $\ mathbb R 1米$ 。该问题的解集（称为 Steiner 紧集）根据到问题的距离分为不同的类 $A_i$ . 每个类包含一个最大包含元素和最小包含元素（分别是最大 Steiner 紧集和最小 Steiner 紧集）。我们找到了一个紧凑集是给定类中的最小 Steiner 紧凑集的必要和充分条件，提供了一种构造这种紧凑集的算法，并找到了对它们的基数的精确估计。我们还提出了一些极小和极大紧集的几何性质。获得的结果可以显着促进特定问题的解决，这可以通过对称集的著名例子来证明 $\ {A_1，A_2，A_3 \} \子集\ mathbb R ^ 2$ ，所有 Steiner 紧集都是不对称的。由于所开发的技术，对这种情况的分析显着简化。

参考书目 16 题。

更新日期：2021-03-09

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