Some algebraic and topological properties of g-convex sets defined by M. A. Noor [Appl. Math. Comput. 199, No. 2, 623–630 (2008; Zbl 1147.65047)] are investigated, in order to deduce their shape. They are a subclass of star-shaped sets, which have also a E. A. Youness-type convexity [J. Optimization Theory Appl. 102, No. 2, 439–450 (1999; Zbl 0937.90082)]. A representation theorem based on
... [Show full abstract] extremal points is given for the class of bounded g-convex sets. Examples showing that this convexity is a very frequent property in connection with a wild range of applications are given.