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

This article is cited in 13 scientific papers (total in 13 papers)

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

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

Bibliographic databases:

UDC: 517
Received: 03.09.2010

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
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\paper Properties of Sets Admitting Stable $\varepsilon$-Selections
\jour Mat. Zametki
\yr 2011
\vol 89
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\pages 572--576
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet
    2. I. G. Tsar'kov, “Continuous $\varepsilon$-selection”, Sb. Math., 207:2 (2016), 267–285  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. I. G. Tsar'kov, “Local and global continuous $\varepsilon$-selection”, Izv. Math., 80:2 (2016), 442–461  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. I. G. Tsar'kov, “Continuous selection for set-valued mappings”, Izv. Math., 81:3 (2017), 645–669  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Tsar'kov I.G., “Continuous Selection From the Sets of Best and Near-Best Approximation”, Dokl. Math., 96:1 (2017), 362–364  crossref  mathscinet  zmath  isi  scopus
    8. I. G. Tsar'kov, “Continuous selections for metric projection operators and for their generalizations”, Izv. Math., 82:4 (2018), 837–859  mathnet  crossref  crossref  adsnasa  isi  elib
    9. I. G. Tsar'kov, “Continuous selections in asymmetric spaces”, Sb. Math., 209:4 (2018), 560–579  mathnet  crossref  crossref  adsnasa  isi  elib
    10. I. G. Tsar'kov, “New Criteria for the Existence of a Continuous $\varepsilon$-Selection”, Math. Notes, 104:5 (2018), 727–734  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref  crossref  isi  elib
    12. I. G. Tsar'kov, “Local Approximation Properties of Sets and Continuous Selections on Them”, Math. Notes, 106:6 (2019), 995–1008  mathnet  crossref  crossref  isi
    13. I. G. Tsar'kov, “Weakly monotone sets and continuous selection in asymmetric spaces”, Sb. Math., 210:9 (2019), 1326–1347  mathnet  crossref  crossref  adsnasa
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