Generalized compactness in~linear spaces and its applications

V. Yu. Protasov ; M. E. Shirokov

Sbornik. Mathematics, Tome 200 (2009) no. 5, pp. 697-722

Résumé

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.

Détail

Keywords: $\mu$-compact set, convex hull of a function, stability of a convex set.
Mots-clés : barycentre map