Generalized compactness in linear spaces and its applicationsстатья
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Дата последнего поиска статьи во внешних источниках: 18 июля 2013 г.
Аннотация:The problems of continuity for convex hulls of continuous concave
functions (CE-property) and for convex hulls of arbitrary convex
functions (strong CE-property) arise naturally for various convex
domains in linear metric spaces. In case of compact domains a
comprehensive solution was elaborated in seventies by Vesterstrom
and O'Brien. First Vesterstrom showed that for compact sets
CE-property is equivalent to the openness of the barycenter map,
while the strong CE-property is equivalent to the openness of the
restriction of that map to the set of maximal measures. Then
O'Brien proved that those two properties are actually both
equivalent to a geometrically obvious ``stability property'' of
convex compacts. This yields, in particular, the equivalency of
the CE-property to the strong CE-property for convex compact sets.
In this paper we give a solution to the following problem: whether
those results can be extended to non-compact 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 class should be the maximal one, for which such
extensions are possible. Then properties of $\mu$-compact sets are
analyzed in detail, several examples are considered, and
applications of the obtained results to the quantum information
theory are discussed.