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

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

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
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
3. A. R. Alimov, “Local solarity of suns in normed linear spaces”, J. Math. Sci., 197:4 (2014), 447–454
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
5. A. R. Alimov, “On finite-dimensional Banach spaces in which suns are connected”, Eurasian Math. J., 6:4 (2015), 7–18
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
7. I. G. Tsar'kov, “Continuous $\varepsilon$-Selection and Monotone Path-Connected Sets”, Math. Notes, 101:6 (2017), 1040–1049
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