RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2011, Volume 89, Issue 4, Pages 608–613 (Mi mz9101)

Properties of Sets Admitting Stable $\varepsilon$-Selections

I. G. Tsar'kov

M. V. Lomonosov Moscow State University

Abstract: Sets in which some convex subsets admit local (global) continuous $\varepsilon$-selections are studied. In particular, it is shown that if, for any number $\varepsilon>0$, some neighborhood $O(x)$ of a point $x$ in a Banach space $X$ contains a dense (in $O(x)$) convex set $K$ admitting an upper semicontinuous acyclic (in particular, continuous single-valued) $\varepsilon$-selection to an approximatively compact set $M\subset X$, then $x$ is a $\delta$-sun point; if, in addition, $X\in (R)$, then the set of all points nearest to $x$ in $M$ is a singleton.

Keywords: Banach space, acyclic upper semicontinuous $\varepsilon$-selection, approximatively compact set, $\delta$-sun, convex set

DOI: https://doi.org/10.4213/mzm9101

Full text: PDF file (418 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2011, 89:4, 572–576

Bibliographic databases:

UDC: 517

Citation: I. G. Tsar'kov, “Properties of Sets Admitting Stable $\varepsilon$-Selections”, Mat. Zametki, 89:4 (2011), 608–613; Math. Notes, 89:4 (2011), 572–576

Citation in format AMSBIB
\Bibitem{Tsa11}
\by I.~G.~Tsar'kov
\paper Properties of Sets Admitting Stable $\varepsilon$-Selections
\jour Mat. Zametki
\yr 2011
\vol 89
\issue 4
\pages 608--613
\mathnet{http://mi.mathnet.ru/mz9101}
\crossref{https://doi.org/10.4213/mzm9101}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2856752}
\transl
\jour Math. Notes
\yr 2011
\vol 89
\issue 4
\pages 572--576
\crossref{https://doi.org/10.1134/S0001434611030291}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79955588062}

• http://mi.mathnet.ru/eng/mz9101
• https://doi.org/10.4213/mzm9101
• http://mi.mathnet.ru/eng/mz/v89/i4/p608

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730
2. I. G. Tsar'kov, “Continuous $\varepsilon$-selection”, Sb. Math., 207:2 (2016), 267–285
3. A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77
4. I. G. Tsar'kov, “Local and global continuous $\varepsilon$-selection”, Izv. Math., 80:2 (2016), 442–461
5. I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669
6. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049
7. Tsar'kov I.G., “Continuous Selection From the Sets of Best and Near-Best Approximation”, Dokl. Math., 96:1 (2017), 362–364
8. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859
9. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579
10. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734
11. I. G. Tsar'kov, “Weakly monotone sets and continuous selection from a near-best approximation operator”, Proc. Steklov Inst. Math., 303 (2018), 227–238
12. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008
13. I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347
•  Number of views: This page: 379 Full text: 84 References: 57 First page: 25