On the support of midconvex operators |
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Authors: | Kazimierz Nikodem |
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Affiliation: | (1) Department of Mathematics, Technical University of Lód , Brench in Bielsko-Bia a, PL-43-309 Bielsko-Bia a, Poland |
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Abstract: | Summary LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D Y is midconvex,g: D Y {– } is midconcave andg f onD, then there exists a Jensen mappingh: D Y {– } such thatg h f onD. Using this theorem we show that a mappingf: D Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth |
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Keywords: | Primary 26A51 secondary 39B70, 46A40 |
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