Certain Subsets on Which Every Bounded Convex Function Is Continuous  

作　　者：Li Xin CHENG Yan Mei TENG 

机构地区：[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

出　　处：《Acta Mathematica Sinica,English Series》2007年第6期1063-1066,共4页数学学报（英文版）

基　　金：the National Natural Science Fundation of China(Grant Nos.10571011,10601005,10471114)

摘　　要：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 C 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.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 C 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.

关 键 词：Convex function BOUNDEDNESS CONTINUITY Banach space 

分 类 号：O17[理学—数学]