Back to Search
Start Over
Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces.
- Source :
- Journal of Optimization Theory & Applications; Sep2005, Vol. 126 Issue 3, p685-700, 16p
- Publication Year :
- 2005
-
Abstract
- For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality f(x′<subscript>0</subscript>) + f (x′<subscript>1</subscript>) ⩽ f(x<subscript>0</subscript>) + f(x<subscript>1</subscript>), for x′<subscript>i</subscript> ∊ [x<subscript>0</subscript>, x<subscript>1</subscript>], for x′<subscript>i</subscript> ∊[x<subscript>0</subscript>, x<subscript>1</subscript>], ∥x′<subscript>i</subscript> - x<subscript>i</subscript>∥ = γ, i = 0, 1, whenever x<subscript>0</subscript>, x<subscript>1</subscript> ∊ D and ∥x<subscript>0</subscript> - x<subscript>1</subscript>∥ ➮ γ. This paper presents some results on the boundedness and continuity of γ-convex functions. For instance, (a) if there is some x<subscript>*</subscript> ∊ D such that f is bounded below on D ∩ B (x<subscript>*</subscript>, γ), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball B(x<subscript>*</subscript>, γ/2) ⊂ D and D′ is a closed bounded subset of D, then f is bounded on D′ if it is bounded above on the boundary of D′; (c) if dim X > 1 and the interior of D contains a closed ball of radius γ, then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x<subscript>*</subscript>, γ/2) in which f has at most countably many discontinuities, then the set of all points at which f is continuous is dense in D. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00223239
- Volume :
- 126
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Journal of Optimization Theory & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 18371456
- Full Text :
- https://doi.org/10.1007/s10957-005-5503-7