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 Mat. Sb., 2009, Volume 200, Number 5, Pages 71–98 (Mi msb5246)

Generalized compactness in linear spaces and its applications

V. Yu. Protasova, M. E. Shirokovb

a Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: For a fixed convex domain in a linear metric space the problems of the continuity of convex envelopes (hulls) of continuous concave functions (the CE-property) and of convex envelopes (hulls) of arbitrary continuous functions (the strong CE-property) arise naturally. In the case of compact domains a comprehensive solution was elaborated in the 1970s by Vesterstrom and O'Brien. First Vesterstrom showed that for compact sets the strong CE-property is equivalent to the openness of the barycentre map, while the CE-property is equivalent to the openness of the restriction of this map to the set of maximal measures. Then O'Brien proved that in fact both properties are equivalent to a geometrically obvious ‘stability property’ of convex compact sets. This yields, in particular, the equivalence of the CE-property to the strong CE-property for convex compact sets. In this paper we give a solution to the following problem: can these results be extended to noncompact convex sets, and, if the answer is positive, to which sets? We show that such an extension does exist. This is an extension to the class of so-called $\mu$-compact sets. Moreover, certain arguments confirm that this could be the maximal class to which such extensions are possible. Then properties of $\mu$-compact sets are analysed in detail, several examples are considered, and applications of the results obtained to quantum information theory are discussed.
Bibliography: 32 titles.

Keywords: barycentre map, $\mu$-compact set, convex hull of a function, stability of a convex set.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm5246

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English version:
Sbornik: Mathematics, 2009, 200:5, 697–722

Bibliographic databases:

UDC: 517.982.25
MSC: Primary 46A50, 46A55; Secondary 47N50

Citation: V. Yu. Protasov, M. E. Shirokov, “Generalized compactness in linear spaces and its applications”, Mat. Sb., 200:5 (2009), 71–98; Sb. Math., 200:5 (2009), 697–722

Citation in format AMSBIB
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• https://doi.org/10.4213/sm5246
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This publication is cited in the following articles:
1. Shirokov M.E., “Continuity of the von Neumann Entropy”, Comm. Math. Phys., 296:3 (2010), 625–654
2. 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
3. M. E. Shirokov, “Stability of convex sets and applications”, Izv. Math., 76:4 (2012), 840–856
4. W. Stephan, “Continuity of the maximum-entropy inference”, Comm. Math. Phys., 330:3 (2014), 1263–1292
5. Geng, Yanlin, Nair Chandra, “The capacity region of the two-receiver Gaussian vector broadcast channel with private and common messages”, IEEE Trans. Inform. Theory, 60:4 (2014), 2087–2104
6. A. S. Holevo, M. E. Shirokov, “Criterion of weak compactness for families of generalized quantum ensembles and its applications”, Theory Probab. Appl., 60:2 (2016), 320–325
7. Weis S., “Maximum-Entropy Inference and Inverse Continuity of the Numerical Range”, Rep. Math. Phys., 77:2 (2016), 251–263
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