Geodesic
On the Strong CE-Property of Convex Sets
Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 441-458.

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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. 
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M. E. Shirokov. On the Strong CE-Property of Convex Sets. Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 441-458. http://geodesic.mathdoc.fr/item/MZM_2007_82_3_a11/

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