Certain Subsets on Which Every Bounded Convex Function Is Continuous |
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Authors: | Li Xin Cheng Yan Mei Teng |
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Institution: | (1) School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China;(2) Department of Mathematics, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China |
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Abstract: | To guarantee every real-valued convex function bounded above on a set is continuous, how ”thick” should the set be? For a symmetric set A in a Banach space E, the answer of this paper is: Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold: i) spanA has finite co-dimentions and ii) coA has nonempty relative interior. This paper also shows that a subset A ? E satisfying every real-valued convex function bounded above on A is continuous on E if (and only if) every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E. |
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Keywords: | Convex function Boundedness Continuity Banach space |
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