Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation |
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Authors: | Emil Ernst Constantin Z?linescu |
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Affiliation: | a Laboratoire de Modélisation en Mécanique et Thermodynamique, Faculté de Sciences et Techniques de Saint Jérôme, Case 322, Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20, France b LACO, UPRESA 6090, Université de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France c University Al.I.Cuza, Iasi, Faculty of Mathematics, 700506-Iasi, Romania |
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Abstract: | The concept of continuous set has been used in finite dimension by Gale and Klee and recently by Auslender and Coutat. Here, we introduce the notion of slice-continuous set in a reflexive Banach space and we show that the class of such sets can be viewed as a subclass of the class of continuous sets. Further, we prove that every nonconstant real-valued convex and continuous function, which has a global minima, attains its infimum on every nonempty convex and closed subset of a reflexive Banach space if and only if its nonempty level sets are slice-continuous. Thereafter, we provide a new separation property for closed convex sets, in terms of slice-continuity, and conclude this article by comments. |
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Keywords: | 47H05 52A41 39B82 |
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