IS EVERY RADIANT FUNCTION THE SUM OF QUASICONVEX FUNCTIONS?

An open question in the study of quasiconvex function is the characterization of the class of functions which are sum of quasiconvex functions. In this paper we restrict attention to quasiconvex radiant functions, i.e. those whose level sets are radiant as well as convex and deal with the claim that a function can be expressed as the sum of quasiconvex radiant functions if and only if it is radiant. Our study is carried out in the framework of Abstract Convex Analysis: the main tool is the description of a supremal generator of the set of radiant functions, i.e. a class of elementary functions whose sup-envelope gives radiant functions, and of the relation between the elementary generators of radiant functions and those of quasiconvex radiant functions. An important intermediate result is a nonlinear separation theorem in which a superlinear function is used to separate a point from a closed radiant set.