Fermat-Steiner Problem in the Metric Space of Compact Sets endowed with Hausdorff Distance
Autor: | Ivanov, Alexandr, Tropin, Alexandr, Tuzhilin, Alexey |
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Rok vydání: | 2016 |
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Druh dokumentu: | Working Paper |
Popis: | The Fermat-Steiner problem consists in finding all points in a metric space $Y$ such that the sum of distances from each of them to the points from some fixed finite subset of $Y$ is minimal. This problem is investigated for the metric space $Y=H(X)$ of compact subsets of a metric space $X$, endowed with the Hausdorff distance. For the case of a proper metric space $X$ a description of all compacts $K\in H(X)$ which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct an example of a regular triangle in $H(R^2)$, such that all its shortest trees have no "natural" symmetry. Comment: 15 pages, 5 Figures |
Databáze: | arXiv |
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