Let f \(*(\epsilon)=\inf \{f(x,\epsilon)|\) \(x\in R(\epsilon)\}\), where f: E \(n\times E\) \(k\to E\) 1 and R: E \(k\to {\mathfrak P}(E\) n). In a previous paper, the authors [J. Optimization Theory Appl. 48, 95-126 (1986; Zbl 0556.90085)] studied the convexity and concavity properties of f *. In this new article, several sufficient conditions for the G-convexity and G-concavity of f *, under different assumptions on f and R, are presented. The notions of G-convexity and G-concavity, introduced here, depend on a function G of four real arguments. It is shown that, for suitable choices of G, G-convexity becomes convexity, quasi-convexity or paraconvexity, among other classes of generalized convex functions (and analogously for G-concavity). The particular case when R corresponds to the feasible region of a parametric inequality-equality constrained nonlinear programming problem is examined. Moreover, a diagram showing all the relationships between the classes of point-to-set maps considered in the paper is provided.