62G30,62H10

In this paper we summarize the known results and the main tools concerning complete systems of inequalities for families of convex sets. We discuss also the possibility to use these systems to determine particular subfamilies of planar convex sets with specific geometric significance. We also analyze complete systems of inequalities for 3-rotationally symmetric planar convex sets concerning the area, the perimeter, the circumradius, the inradius, the diameter and the minimal width; we give a list of new inequalities concerning these parameters and we point out which are the cases that are still open.;