1. Certain Subsets on Which Every Bounded Convex Function Is Continuous.
- Author
-
Li Xin Cheng and Yan Mei Teng
- Subjects
- *
CONVEX functions , *REAL variables , *CONTINUITY , *BANACH spaces , *DISTANCE geometry , *SUBDIFFERENTIALS , *BAIRE spaces , *COMPLEX variables - 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) co A 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. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF