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Sbornik: Mathematics, 2020, Volume 211, Issue 10, Pages 1382–1398
DOI: https://doi.org/10.1070/SM9361
(Mi sm9361)
 

This article is cited in 1 scientific paper (total in 1 paper)

Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric

O. S. Malysheva

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
References:
Abstract: We study the geometry of the metric space of compact subsets of $\mathbb R^n$ considered up to an orientation-preserving motion. We show that, in the optimal position of a pair of compact sets (for which the Hausdorff distance between the sets cannot be decreased), one of which is a singleton, this point is at the Chebyshev centre of the other. For orientedly similar compacta we evaluate the Euclidean Gromov-Hausdorff distance between them and prove that, in the optimal position, the Chebyshev centres of these compacta coincide. We show that every three-point metric space can be embedded isometrically in the space of compacta under consideration. We prove that, for a pair of optimally positioned compacta all compacta that lie in between in the sense of the Hausdorff metric also lie in between in the sense of the Euclidean Gromov-Hausdorff metric. For an arbitrary $n$-point boundary formed by compact sets of a set $\mathscr X$ that are neighbourhoods of segments, the Steiner point realizes the minimal filling and also belongs to the set $\mathscr X$.
Bibliography: 14 titles.
Keywords: Steiner point, Euclidean Gromov-Hausdorff metric, optimal position of compacta.
Funding agency Grant number
Russian Foundation for Basic Research 19-01-00775-а
This research was supported by the Russian Foundation for Basic Research (grant no. 19-01-00775-a).
Received: 11.12.2019 and 17.04.2020
Bibliographic databases:
Document Type: Article
UDC: 515.124+514.177.2
MSC: Primary 51F99; Secondary 51K05
Language: English
Original paper language: Russian
Citation: O. S. Malysheva, “Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric”, Sb. Math., 211:10 (2020), 1382–1398
Citation in format AMSBIB
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\by O.~S.~Malysheva
\paper Optimal position of compact sets and the Steiner problem in spaces with Euclidean Gromov-Hausdorff metric
\jour Sb. Math.
\yr 2020
\vol 211
\issue 10
\pages 1382--1398
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\crossref{https://doi.org/10.1070/SM9361}
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Linking options:
  • https://www.mathnet.ru/eng/sm9361
  • https://doi.org/10.1070/SM9361
  • https://www.mathnet.ru/eng/sm/v211/i10/p32
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    References:33
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