This talk deals with error bound characterizations of the conical constraint qualification (CCQ) for convex inequality systems in a Banach space. We establish necessary and sufficient conditions for a closed convex set defined by a convex function to have CCQ. We show that these characterizations only take place in certain very specific situations.

We give technical examples showing that these characterizations are limited to these ones. We introduce a new condition in terms of the gauge function which allows us to give an error bound characterization of convex nondifferentiable systems and to obtain as a direct consequence different characterizations of the concept of the strong conical hull intersection property (CHIP) for a finite collection of convex sets.

This is based on the joint work with A. Barbara.