
This article is cited in 7 scientific papers (total in 7 papers)
Generalized compactness in linear spaces and its applications
V. Yu. Protasov^{a}, M. E. Shirokov^{b}^{†} ^{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 CEproperty) and of convex envelopes (hulls) of arbitrary continuous functions (the strong CEproperty) 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 CEproperty is equivalent to the openness of the barycentre map, while the CEproperty 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 CEproperty to the strong CEproperty 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 socalled $\mu$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 Received: 09.04.2008 and 17.02.2009
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
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W. Stephan, “Continuity of the maximumentropy inference”, Comm. Math. Phys., 330:3 (2014), 1263–1292

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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

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