Section

Mathematics

Title

On one metric in the space of nonempty closed subsets of $\mathbb{R}^n$

Author(s)

Zhukovskiy E.S.^{a},
Panasenko E.A.^{a}

Affiliations

Tambov State University^{a}

Abstract

In the work, there is presented a new metric in the space ${\rm clos}(\mathbb{R}^n)$ of all nonempty closed (not necessarily bounded) subsets of $\mathbb{R}^n$. The convergence of sets in this metric is equivalent to convergence in the Hausdorff metric of the intersections of the given sets with the balls of any positive radius centered at zero united then with the corresponding spheres. It is proved that, with respect to the metric considered, the space ${\rm clos}(\mathbb{R}^n)$ is complete, and its subspace of nonempty closed convex subsets of $\mathbb{R}^n$ is closed. There are also derived the conditions that guarantee the equivalence of convergence in this metric to convergence in the Hausdorff metric, and to convergence in the Hausdorff–Bebutov metric. The results obtained can be applied to studying control problems and differential inclusions.

Keywords

complete metric space of nonempty closed subsets of $\mathbb{R}^n$, subspaces, convergence

UDC

515.124, 517.911.5

MSC

54E50, 34A60

DOI

10.20537/vm120102

Received

12 October 2011

Language

Russian

Citation

Zhukovskiy E.S., Panasenko E.A. On one metric in the space of nonempty closed subsets of $\mathbb{R}^n$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 1, pp. 1525.

References

 Panasenko E.A., Tonkov E.L. Extension of E.A. Barbashin’s and N.N. Krasovskii’s stability theorems to controlled dynamical systems, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 268, suppl. 1, pp. 204–221.
 Panasenko E.A., Rodina L.I., Tonkov E.L. The space ${\rm clcv}($ $\mathbb{R}$^{$n$} $)$ with the Hausdorff–Bebutov metric and differential inclusions, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 275, suppl. 1, pp. 121–136.
 Leichtweiss K. Vypuklye mnozhestva (Convex sets), Moscow: Nauka, 1985, 335 p.

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