A Systematization of Convexity Concepts for Sets and Functions

The paper surveys several convexity concepts, referring to sets and to functions respectively, with the purpose to put them in some kind of order according to the same principle. First it examines the connections between six convexity concepts regarding sets in topological linear spaces and points out the most general concept among these concepts. On the basis of this analysis it is then revealed that twelve convexity concepts concerning functions, that take values in topological linear spaces, can be naturally defined by reduction to the investigated convexities for sets. The most general convexity concept for functions is also found. It is applied to establish an alternative theorem as well as necessary optimality conditions for weak multiobjective optimization problems.