Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

Abstract

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature

Keywords:

• Convex Functions
• Generalized Convexity
• Guasiconvexity
• Midpoint Convexity