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 Mat. Zametki, 2007, Volume 82, Issue 3, Pages 441–458 (Mi mz3845)

On the Strong CE-Property of Convex Sets

M. E. Shirokov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We consider a class of convex bounded subsets of a separable Banach space. This class includes all convex compact sets as well as some noncompact sets important in applications. For sets in this class, we obtain a simple criterion for the strong CE-property, i.e., the property that the convex closure of any continuous bounded function is a continuous bounded function. Some results are obtained concerning the extension of functions defined at the extreme points of a set in this class to convex or concave functions defined on the entire set with preservation of closedness and continuity. Some applications of the results in quantum information theory are considered.

Keywords: compact set, continuity, convex function, concave function, convex envelope, convex closure, $\mathrm{CE}$-property, topological linear space, separable Banach space

DOI: https://doi.org/10.4213/mzm3845

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English version:
Mathematical Notes, 2007, 82:3, 395–409

Bibliographic databases:

UDC: 517.982
Revised: 22.02.2007

Citation: M. E. Shirokov, “On the Strong CE-Property of Convex Sets”, Mat. Zametki, 82:3 (2007), 441–458; Math. Notes, 82:3 (2007), 395–409

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz3845
• https://doi.org/10.4213/mzm3845
• http://mi.mathnet.ru/eng/mz/v82/i3/p441

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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. M. E. Shirokov, “Characterization of convex $\mu$-compact sets”, Russian Math. Surveys, 63:5 (2008), 981–982
2. V. Yu. Protasov, M. E. Shirokov, “Generalized compactness in linear spaces and its applications”, Sb. Math., 200:5 (2009), 697–722
3. M. E. Shirokov, “On properties of the space of quantum states and their application to the construction of entanglement monotones”, Izv. Math., 74:4 (2010), 849–882
4. M. I. Gomoyunov, N. Yu. Lukoyanov, “On the stability of a procedure for solving a minimax control problem for a positional functional”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 54–69
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