Generalized convexities of lower semicontinuous functions

Abstract

In this paper a general theorem on the replacement of the condition “for all λ” in the definition of generalized convexity properties of lower semicontinuous functions by the condition “there exists a λ” is shown. This result can be applied to a number of special kinds of convexity and completes, for instance, studies of Behbikgeb concerning (explicitly) quasiconvex functions.

Keywords:

• Generalized convex functions
• convexlikeness
• quasiconvexity
• explicit quasi-convexity

AMS 1980 Subject Classifications:

• Primary: 26 B 25
• Secondary: 90 C 25