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Fundam. Prikl. Mat., 2012, Volume 17, Issue 4, Pages 3–12 (Mi fpm1418)  

This article is cited in 7 scientific papers (total in 7 papers)

Separation of convex sets by extreme hyperplanes

A. R. Alimov, V. Yu. Protasov

M. V. Lomonosov Moscow State University

Abstract: The problem of separation of convex sets by extreme hyperplanes (functionals) in normed linear spaces is examined. A concept of a bar (a closed set of a special form) is introduced; it is shown that a bar is characterized by the property that any point not lying in it can be separated from it by an extreme hyperplane. In two-dimensional spaces, in spaces with strictly convex dual, and in the space of continuous functions, any two bars are extremely separated. This property is shown to fail in the space of summable functions. A number of examples and generalizations are given.

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English version:
Journal of Mathematical Sciences (New York), 2013, 191:5, 599–604

Document Type: Article
UDC: 517.982.252

Citation: A. R. Alimov, V. Yu. Protasov, “Separation of convex sets by extreme hyperplanes”, Fundam. Prikl. Mat., 17:4 (2012), 3–12; J. Math. Sci., 191:5 (2013), 599–604

Citation in format AMSBIB
\Bibitem{AliPro12}
\by A.~R.~Alimov, V.~Yu.~Protasov
\paper Separation of convex sets by extreme hyperplanes
\jour Fundam. Prikl. Mat.
\yr 2012
\vol 17
\issue 4
\pages 3--12
\mathnet{http://mi.mathnet.ru/fpm1418}
\transl
\jour J. Math. Sci.
\yr 2013
\vol 191
\issue 5
\pages 599--604
\crossref{https://doi.org/10.1007/s10958-013-1345-2}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884983149}


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    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, “Monotone path-connectedness of $R$-weakly convex sets in spaces with linear ball embedding”, Eurasian Math. J., 3:2 (2012), 21–30  mathnet  mathscinet  zmath
    2. Alimov A.R., “Ogranichennaya strogaya solnechnost strogikh solnts v prostranstve c(q)”, Vestnik moskovskogo universiteta. seriya 1: matematika. mekhanika, 2012, no. 6, 16–19  mathnet  mathscinet  elib
    3. A. R. Alimov, “Local solarity of suns in normed linear spaces”, J. Math. Sci., 197:4 (2014), 447–454  mathnet  crossref
    4. 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
    5. A. R. Alimov, “On finite-dimensional Banach spaces in which suns are connected”, Eurasian Math. J., 6:4 (2015), 7–18  mathnet
    6. 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
    7. 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
  • Фундаментальная и прикладная математика
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