|
This article is cited in 3 scientific papers (total in 3 papers)
The Fermat-Steiner problem in the space of compact subsets of $\mathbb R^m$ endowed with the Hausdorff metric
A. Kh. Galstyanab, A. O. Ivanovabc, A. A. Tuzhilinab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics
c Bauman Moscow State Technical University
Abstract:
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.
Keywords:
minimal networks, Hausdorff distance, Fermat-Steiner problem, Steiner problem, metric geometry.
Received: 02.11.2019
Citation:
A. Kh. Galstyan, A. O. Ivanov, A. A. Tuzhilin, “The Fermat-Steiner problem in the space of compact subsets of $\mathbb R^m$ endowed with the Hausdorff metric”, Mat. Sb., 212:1 (2021), 28–62; Sb. Math., 212:1 (2021), 25–56
Linking options:
https://www.mathnet.ru/eng/sm9343https://doi.org/10.1070/SM9343 https://www.mathnet.ru/eng/sm/v212/i1/p28
|
Statistics & downloads: |
Abstract page: | 391 | Russian version PDF: | 68 | English version PDF: | 18 | Russian version HTML: | 127 | References: | 23 | First page: | 13 |
|